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G = C7×C22.34C24  order 448 = 26·7

Direct product of C7 and C22.34C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C22.34C24, C14.1562+ (1+4), (C4×D4)⋊12C14, (D4×C28)⋊41C2, C4⋊D4⋊9C14, C4⋊1D4⋊6C14, C42.C2⋊5C14, C42.38(C2×C14), C42⋊C2⋊12C14, C28.276(C4○D4), (C4×C28).279C22, (C2×C14).360C24, (C2×C28).669C23, C22.D4⋊6C14, C2.8(C7×2+ (1+4)), (D4×C14).218C22, C23.12(C22×C14), (C22×C14).95C23, C22.34(C23×C14), (C22×C28).448C22, C4.20(C7×C4○D4), (C7×C4⋊1D4)⋊17C2, C4⋊C4.29(C2×C14), (C7×C4⋊D4)⋊36C2, C2.17(C14×C4○D4), (C2×D4).32(C2×C14), C14.236(C2×C4○D4), (C7×C42.C2)⋊22C2, (C7×C42⋊C2)⋊33C2, C22⋊C4.16(C2×C14), (C7×C4⋊C4).393C22, (C22×C4).60(C2×C14), (C2×C4).27(C22×C14), (C7×C22.D4)⋊25C2, (C7×C22⋊C4).86C22, SmallGroup(448,1323)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.34C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — C7×C22⋊C4 — C7×C4⋊D4 — C7×C22.34C24
Lower central
C1 — C22 — C7×C22.34C24
Upper central
C1 — C2×C14 — C7×C22.34C24

Subgroups: 402 in 240 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C7, C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×12], C23, C23 [×4], C14, C14 [×2], C14 [×5], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×10], C28 [×2], C28 [×9], C2×C14, C2×C14 [×15], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C2×C28 [×2], C2×C28 [×8], C2×C28 [×6], C7×D4 [×12], C22×C14, C22×C14 [×4], C22.34C24, C4×C28 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×8], C22×C28, C22×C28 [×4], D4×C14 [×10], C7×C42⋊C2, D4×C28 [×2], C7×C4⋊D4 [×6], C7×C22.D4 [×4], C7×C42.C2, C7×C4⋊1D4, C7×C22.34C24

Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C4○D4 [×2], C24, C2×C14 [×35], C2×C4○D4, 2+ (1+4) [×2], C22×C14 [×15], C22.34C24, C7×C4○D4 [×2], C23×C14, C14×C4○D4, C7×2+ (1+4) [×2], C7×C22.34C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=f2=1, e2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Smallest permutation representation
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)

(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 127)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 211)

(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(36 47)(37 48)(38 49)(39 43)(40 44)(41 45)(42 46)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 123)(114 124)(115 125)(116 126)(117 120)(118 121)(119 122)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(169 179)(170 180)(171 181)(172 182)(173 176)(174 177)(175 178)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)

(1 206)(2 207)(3 208)(4 209)(5 210)(6 204)(7 205)(8 77)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 61)(23 62)(24 63)(25 57)(26 58)(27 59)(28 60)(29 197)(30 198)(31 199)(32 200)(33 201)(34 202)(35 203)(36 194)(37 195)(38 196)(39 190)(40 191)(41 192)(42 193)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(64 223)(65 224)(66 218)(67 219)(68 220)(69 221)(70 222)(78 147)(79 141)(80 142)(81 143)(82 144)(83 145)(84 146)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 134)(92 160)(93 161)(94 155)(95 156)(96 157)(97 158)(98 159)(99 150)(100 151)(101 152)(102 153)(103 154)(104 148)(105 149)(106 183)(107 184)(108 185)(109 186)(110 187)(111 188)(112 189)(113 179)(114 180)(115 181)(116 182)(117 176)(118 177)(119 178)(120 173)(121 174)(122 175)(123 169)(124 170)(125 171)(126 172)(127 162)(128 163)(129 164)(130 165)(131 166)(132 167)(133 168)

(1 147 35 150)(2 141 29 151)(3 142 30 152)(4 143 31 153)(5 144 32 154)(6 145 33 148)(7 146 34 149)(8 118 21 121)(9 119 15 122)(10 113 16 123)(11 114 17 124)(12 115 18 125)(13 116 19 126)(14 117 20 120)(22 132 223 111)(23 133 224 112)(24 127 218 106)(25 128 219 107)(26 129 220 108)(27 130 221 109)(28 131 222 110)(36 159 47 138)(37 160 48 139)(38 161 49 140)(39 155 43 134)(40 156 44 135)(41 157 45 136)(42 158 46 137)(50 183 71 162)(51 184 72 163)(52 185 73 164)(53 186 74 165)(54 187 75 166)(55 188 76 167)(56 189 77 168)(57 179 67 169)(58 180 68 170)(59 181 69 171)(60 182 70 172)(61 176 64 173)(62 177 65 174)(63 178 66 175)(78 211 99 190)(79 212 100 191)(80 213 101 192)(81 214 102 193)(82 215 103 194)(83 216 104 195)(84 217 105 196)(85 207 95 197)(86 208 96 198)(87 209 97 199)(88 210 98 200)(89 204 92 201)(90 205 93 202)(91 206 94 203)

(1 94)(2 95)(3 96)(4 97)(5 98)(6 92)(7 93)(8 189)(9 183)(10 184)(11 185)(12 186)(13 187)(14 188)(15 162)(16 163)(17 164)(18 165)(19 166)(20 167)(21 168)(22 173)(23 174)(24 175)(25 169)(26 170)(27 171)(28 172)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 82)(37 83)(38 84)(39 78)(40 79)(41 80)(42 81)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(134 206)(135 207)(136 208)(137 209)(138 210)(139 204)(140 205)(141 212)(142 213)(143 214)(144 215)(145 216)(146 217)(147 211)(148 195)(149 196)(150 190)(151 191)(152 192)(153 193)(154 194)(155 203)(156 197)(157 198)(158 199)(159 200)(160 201)(161 202)(176 223)(177 224)(178 218)(179 219)(180 220)(181 221)(182 222)

(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 202 23 217)(9 203 24 211)(10 197 25 212)(11 198 26 213)(12 199 27 214)(13 200 28 215)(14 201 22 216)(15 206 218 190)(16 207 219 191)(17 208 220 192)(18 209 221 193)(19 210 222 194)(20 204 223 195)(21 205 224 196)(29 72 44 57)(30 73 45 58)(31 74 46 59)(32 75 47 60)(33 76 48 61)(34 77 49 62)(35 71 43 63)(78 122 94 106)(79 123 95 107)(80 124 96 108)(81 125 97 109)(82 126 98 110)(83 120 92 111)(84 121 93 112)(85 128 100 113)(86 129 101 114)(87 130 102 115)(88 131 103 116)(89 132 104 117)(90 133 105 118)(91 127 99 119)(134 178 150 162)(135 179 151 163)(136 180 152 164)(137 181 153 165)(138 182 154 166)(139 176 148 167)(140 177 149 168)(141 184 156 169)(142 185 157 170)(143 186 158 171)(144 187 159 172)(145 188 160 173)(146 189 161 174)(147 183 155 175)
 
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,206)(2,207)(3,208)(4,209)(5,210)(6,204)(7,205)(8,77)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,61)(23,62)(24,63)(25,57)(26,58)(27,59)(28,60)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,194)(37,195)(38,196)(39,190)(40,191)(41,192)(42,193)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(64,223)(65,224)(66,218)(67,219)(68,220)(69,221)(70,222)(78,147)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,134)(92,160)(93,161)(94,155)(95,156)(96,157)(97,158)(98,159)(99,150)(100,151)(101,152)(102,153)(103,154)(104,148)(105,149)(106,183)(107,184)(108,185)(109,186)(110,187)(111,188)(112,189)(113,179)(114,180)(115,181)(116,182)(117,176)(118,177)(119,178)(120,173)(121,174)(122,175)(123,169)(124,170)(125,171)(126,172)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168), (1,147,35,150)(2,141,29,151)(3,142,30,152)(4,143,31,153)(5,144,32,154)(6,145,33,148)(7,146,34,149)(8,118,21,121)(9,119,15,122)(10,113,16,123)(11,114,17,124)(12,115,18,125)(13,116,19,126)(14,117,20,120)(22,132,223,111)(23,133,224,112)(24,127,218,106)(25,128,219,107)(26,129,220,108)(27,130,221,109)(28,131,222,110)(36,159,47,138)(37,160,48,139)(38,161,49,140)(39,155,43,134)(40,156,44,135)(41,157,45,136)(42,158,46,137)(50,183,71,162)(51,184,72,163)(52,185,73,164)(53,186,74,165)(54,187,75,166)(55,188,76,167)(56,189,77,168)(57,179,67,169)(58,180,68,170)(59,181,69,171)(60,182,70,172)(61,176,64,173)(62,177,65,174)(63,178,66,175)(78,211,99,190)(79,212,100,191)(80,213,101,192)(81,214,102,193)(82,215,103,194)(83,216,104,195)(84,217,105,196)(85,207,95,197)(86,208,96,198)(87,209,97,199)(88,210,98,200)(89,204,92,201)(90,205,93,202)(91,206,94,203), (1,94)(2,95)(3,96)(4,97)(5,98)(6,92)(7,93)(8,189)(9,183)(10,184)(11,185)(12,186)(13,187)(14,188)(15,162)(16,163)(17,164)(18,165)(19,166)(20,167)(21,168)(22,173)(23,174)(24,175)(25,169)(26,170)(27,171)(28,172)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,82)(37,83)(38,84)(39,78)(40,79)(41,80)(42,81)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(134,206)(135,207)(136,208)(137,209)(138,210)(139,204)(140,205)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,211)(148,195)(149,196)(150,190)(151,191)(152,192)(153,193)(154,194)(155,203)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)(176,223)(177,224)(178,218)(179,219)(180,220)(181,221)(182,222), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,202,23,217)(9,203,24,211)(10,197,25,212)(11,198,26,213)(12,199,27,214)(13,200,28,215)(14,201,22,216)(15,206,218,190)(16,207,219,191)(17,208,220,192)(18,209,221,193)(19,210,222,194)(20,204,223,195)(21,205,224,196)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,128,100,113)(86,129,101,114)(87,130,102,115)(88,131,103,116)(89,132,104,117)(90,133,105,118)(91,127,99,119)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,184,156,169)(142,185,157,170)(143,186,158,171)(144,187,159,172)(145,188,160,173)(146,189,161,174)(147,183,155,175)>;
 
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,206)(2,207)(3,208)(4,209)(5,210)(6,204)(7,205)(8,77)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,61)(23,62)(24,63)(25,57)(26,58)(27,59)(28,60)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,194)(37,195)(38,196)(39,190)(40,191)(41,192)(42,193)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(64,223)(65,224)(66,218)(67,219)(68,220)(69,221)(70,222)(78,147)(79,141)(80,142)(81,143)(82,144)(83,145)(84,146)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,134)(92,160)(93,161)(94,155)(95,156)(96,157)(97,158)(98,159)(99,150)(100,151)(101,152)(102,153)(103,154)(104,148)(105,149)(106,183)(107,184)(108,185)(109,186)(110,187)(111,188)(112,189)(113,179)(114,180)(115,181)(116,182)(117,176)(118,177)(119,178)(120,173)(121,174)(122,175)(123,169)(124,170)(125,171)(126,172)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168), (1,147,35,150)(2,141,29,151)(3,142,30,152)(4,143,31,153)(5,144,32,154)(6,145,33,148)(7,146,34,149)(8,118,21,121)(9,119,15,122)(10,113,16,123)(11,114,17,124)(12,115,18,125)(13,116,19,126)(14,117,20,120)(22,132,223,111)(23,133,224,112)(24,127,218,106)(25,128,219,107)(26,129,220,108)(27,130,221,109)(28,131,222,110)(36,159,47,138)(37,160,48,139)(38,161,49,140)(39,155,43,134)(40,156,44,135)(41,157,45,136)(42,158,46,137)(50,183,71,162)(51,184,72,163)(52,185,73,164)(53,186,74,165)(54,187,75,166)(55,188,76,167)(56,189,77,168)(57,179,67,169)(58,180,68,170)(59,181,69,171)(60,182,70,172)(61,176,64,173)(62,177,65,174)(63,178,66,175)(78,211,99,190)(79,212,100,191)(80,213,101,192)(81,214,102,193)(82,215,103,194)(83,216,104,195)(84,217,105,196)(85,207,95,197)(86,208,96,198)(87,209,97,199)(88,210,98,200)(89,204,92,201)(90,205,93,202)(91,206,94,203), (1,94)(2,95)(3,96)(4,97)(5,98)(6,92)(7,93)(8,189)(9,183)(10,184)(11,185)(12,186)(13,187)(14,188)(15,162)(16,163)(17,164)(18,165)(19,166)(20,167)(21,168)(22,173)(23,174)(24,175)(25,169)(26,170)(27,171)(28,172)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,82)(37,83)(38,84)(39,78)(40,79)(41,80)(42,81)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(134,206)(135,207)(136,208)(137,209)(138,210)(139,204)(140,205)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,211)(148,195)(149,196)(150,190)(151,191)(152,192)(153,193)(154,194)(155,203)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)(176,223)(177,224)(178,218)(179,219)(180,220)(181,221)(182,222), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,202,23,217)(9,203,24,211)(10,197,25,212)(11,198,26,213)(12,199,27,214)(13,200,28,215)(14,201,22,216)(15,206,218,190)(16,207,219,191)(17,208,220,192)(18,209,221,193)(19,210,222,194)(20,204,223,195)(21,205,224,196)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,128,100,113)(86,129,101,114)(87,130,102,115)(88,131,103,116)(89,132,104,117)(90,133,105,118)(91,127,99,119)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,184,156,169)(142,185,157,170)(143,186,158,171)(144,187,159,172)(145,188,160,173)(146,189,161,174)(147,183,155,175) );
 
G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99),(106,122),(107,123),(108,124),(109,125),(110,126),(111,120),(112,121),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,127),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,155),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,183),(190,206),(191,207),(192,208),(193,209),(194,210),(195,204),(196,205),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,211)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(36,47),(37,48),(38,49),(39,43),(40,44),(41,45),(42,46),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,123),(114,124),(115,125),(116,126),(117,120),(118,121),(119,122),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189),(169,179),(170,180),(171,181),(172,182),(173,176),(174,177),(175,178),(190,211),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,206),(2,207),(3,208),(4,209),(5,210),(6,204),(7,205),(8,77),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,61),(23,62),(24,63),(25,57),(26,58),(27,59),(28,60),(29,197),(30,198),(31,199),(32,200),(33,201),(34,202),(35,203),(36,194),(37,195),(38,196),(39,190),(40,191),(41,192),(42,193),(43,211),(44,212),(45,213),(46,214),(47,215),(48,216),(49,217),(64,223),(65,224),(66,218),(67,219),(68,220),(69,221),(70,222),(78,147),(79,141),(80,142),(81,143),(82,144),(83,145),(84,146),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,134),(92,160),(93,161),(94,155),(95,156),(96,157),(97,158),(98,159),(99,150),(100,151),(101,152),(102,153),(103,154),(104,148),(105,149),(106,183),(107,184),(108,185),(109,186),(110,187),(111,188),(112,189),(113,179),(114,180),(115,181),(116,182),(117,176),(118,177),(119,178),(120,173),(121,174),(122,175),(123,169),(124,170),(125,171),(126,172),(127,162),(128,163),(129,164),(130,165),(131,166),(132,167),(133,168)], [(1,147,35,150),(2,141,29,151),(3,142,30,152),(4,143,31,153),(5,144,32,154),(6,145,33,148),(7,146,34,149),(8,118,21,121),(9,119,15,122),(10,113,16,123),(11,114,17,124),(12,115,18,125),(13,116,19,126),(14,117,20,120),(22,132,223,111),(23,133,224,112),(24,127,218,106),(25,128,219,107),(26,129,220,108),(27,130,221,109),(28,131,222,110),(36,159,47,138),(37,160,48,139),(38,161,49,140),(39,155,43,134),(40,156,44,135),(41,157,45,136),(42,158,46,137),(50,183,71,162),(51,184,72,163),(52,185,73,164),(53,186,74,165),(54,187,75,166),(55,188,76,167),(56,189,77,168),(57,179,67,169),(58,180,68,170),(59,181,69,171),(60,182,70,172),(61,176,64,173),(62,177,65,174),(63,178,66,175),(78,211,99,190),(79,212,100,191),(80,213,101,192),(81,214,102,193),(82,215,103,194),(83,216,104,195),(84,217,105,196),(85,207,95,197),(86,208,96,198),(87,209,97,199),(88,210,98,200),(89,204,92,201),(90,205,93,202),(91,206,94,203)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,92),(7,93),(8,189),(9,183),(10,184),(11,185),(12,186),(13,187),(14,188),(15,162),(16,163),(17,164),(18,165),(19,166),(20,167),(21,168),(22,173),(23,174),(24,175),(25,169),(26,170),(27,171),(28,172),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,82),(37,83),(38,84),(39,78),(40,79),(41,80),(42,81),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,129),(74,130),(75,131),(76,132),(77,133),(134,206),(135,207),(136,208),(137,209),(138,210),(139,204),(140,205),(141,212),(142,213),(143,214),(144,215),(145,216),(146,217),(147,211),(148,195),(149,196),(150,190),(151,191),(152,192),(153,193),(154,194),(155,203),(156,197),(157,198),(158,199),(159,200),(160,201),(161,202),(176,223),(177,224),(178,218),(179,219),(180,220),(181,221),(182,222)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,202,23,217),(9,203,24,211),(10,197,25,212),(11,198,26,213),(12,199,27,214),(13,200,28,215),(14,201,22,216),(15,206,218,190),(16,207,219,191),(17,208,220,192),(18,209,221,193),(19,210,222,194),(20,204,223,195),(21,205,224,196),(29,72,44,57),(30,73,45,58),(31,74,46,59),(32,75,47,60),(33,76,48,61),(34,77,49,62),(35,71,43,63),(78,122,94,106),(79,123,95,107),(80,124,96,108),(81,125,97,109),(82,126,98,110),(83,120,92,111),(84,121,93,112),(85,128,100,113),(86,129,101,114),(87,130,102,115),(88,131,103,116),(89,132,104,117),(90,133,105,118),(91,127,99,119),(134,178,150,162),(135,179,151,163),(136,180,152,164),(137,181,153,165),(138,182,154,166),(139,176,148,167),(140,177,149,168),(141,184,156,169),(142,185,157,170),(143,186,158,171),(144,187,159,172),(145,188,160,173),(146,189,161,174),(147,183,155,175)])
 

Matrix representation ►G ⊆ GL6(𝔽29)

100000
010000
0016000
0001600
0000160
0000016
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
0028000
0002800
0000280
0000028
,
22240000
2770000
0016030
00265116
0020130
002132324
,
1700000
0170000
0040190
00616165
00220250
0010242013
,
3270000
4260000
00150270
007091
00250140
0018140
,
2800000
0280000
00202700
0012900
0071401
00114280

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[22,27,0,0,0,0,24,7,0,0,0,0,0,0,16,26,2,2,0,0,0,5,0,13,0,0,3,1,13,23,0,0,0,16,0,24],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,4,6,22,10,0,0,0,16,0,24,0,0,19,16,25,20,0,0,0,5,0,13],[3,4,0,0,0,0,27,26,0,0,0,0,0,0,15,7,25,18,0,0,0,0,0,1,0,0,27,9,14,4,0,0,0,1,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,20,12,7,11,0,0,27,9,14,4,0,0,0,0,0,28,0,0,0,0,1,0] >;
 

154 conjugacy classes

class 1 2A2B2C2D···2H4A···4F4G···4M7A···7F14A···14R14S···14AV28A···28AJ28AK···28BZ
order12222···24···44···47···714···1414···1428···2828···28
size11114···42···24···41···11···14···42···24···4

154 irreducible representations

dim111111111111112244
type++++++++
imageC1C2C2C2C2C2C2C7C14C14C14C14C14C14C4○D4C7×C4○D42+ (1+4)C7×2+ (1+4)
kernelC7×C22.34C24C7×C42⋊C2D4×C28C7×C4⋊D4C7×C22.D4C7×C42.C2C7×C4⋊1D4C22.34C24C42⋊C2C4×D4C4⋊D4C22.D4C42.C2C4⋊1D4C28C4C14C2
# reps11264116612362466424212

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{34}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.34C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1323);
 
// by ID
 
G=gap.SmallGroup(448,1323);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,1227,3363,416]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=f^2=1,e^2=c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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×
⋊
ℤ
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