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G = C7×C22.49C24  order 448 = 26·7

Direct product of C7 and C22.49C24

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

Aliases: C7×C22.49C24, C14.1662+ (1+4), C4⋊Q8⋊16C14, (D4×C28)⋊50C2, (C4×D4)⋊21C14, C4⋊D4⋊16C14, C4.4D4⋊13C14, C42.49(C2×C14), C42⋊C2⋊17C14, C28.326(C4○D4), (C2×C14).375C24, (C2×C28).964C23, (C4×C28).290C22, (D4×C14).222C22, C23.20(C22×C14), C22.49(C23×C14), (Q8×C14).185C22, C2.18(C7×2+ (1+4)), (C22×C28).460C22, (C22×C14).103C23, (C7×C4⋊Q8)⋊37C2, C4.38(C7×C4○D4), C4⋊C4.75(C2×C14), (C7×C4⋊D4)⋊43C2, C2.28(C14×C4○D4), (C2×D4).35(C2×C14), C14.247(C2×C4○D4), (C7×C4.4D4)⋊33C2, (C2×Q8).28(C2×C14), (C7×C42⋊C2)⋊38C2, C22⋊C4.25(C2×C14), (C7×C4⋊C4).409C22, (C2×C4).37(C22×C14), (C22×C4).71(C2×C14), (C7×C22⋊C4).157C22, SmallGroup(448,1338)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.49C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — C7×C22⋊C4 — C7×C4.4D4 — C7×C22.49C24
Lower central
C1 — C22 — C7×C22.49C24
Upper central
C1 — C2×C14 — C7×C22.49C24

Subgroups: 362 in 236 conjugacy classes, 150 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C7, C2×C4, C2×C4 [×10], C2×C4 [×8], D4 [×8], Q8 [×2], C23 [×4], C14, C14 [×2], C14 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C28 [×4], C28 [×9], C2×C14, C2×C14 [×12], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C2×C28, C2×C28 [×10], C2×C28 [×8], C7×D4 [×8], C7×Q8 [×2], C22×C14 [×4], C22.49C24, C4×C28, C4×C28 [×4], C7×C22⋊C4 [×12], C7×C4⋊C4 [×6], C22×C28 [×4], D4×C14 [×6], Q8×C14 [×2], C7×C42⋊C2 [×4], D4×C28 [×2], C7×C4⋊D4 [×4], C7×C4.4D4 [×4], C7×C4⋊Q8, C7×C22.49C24

Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C4○D4 [×4], C24, C2×C14 [×35], C2×C4○D4 [×2], 2+ (1+4), C22×C14 [×15], C22.49C24, C7×C4○D4 [×4], C23×C14, C14×C4○D4 [×2], C7×2+ (1+4), C7×C22.49C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=c, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, 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 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(29 45)(30 46)(31 47)(32 48)(33 49)(34 43)(35 44)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 120)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 149)(142 150)(143 151)(144 152)(145 153)(146 154)(147 148)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 177)(170 178)(171 179)(172 180)(173 181)(174 182)(175 176)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 205)(198 206)(199 207)(200 208)(201 209)(202 210)(203 204)

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

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

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

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

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

Matrix representation ►G ⊆ GL4(𝔽29) generated by

23000
02300
0010
0001
,
28000
02800
0010
0001
,
1000
0100
00280
00028
,
01700
12000
00012
00170
,
1000
02800
00170
00017
,
12000
01200
0001
0010
,
0100
28000
00280
00028
G:=sub<GL(4,GF(29))| [23,0,0,0,0,23,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,12,0,0,17,0,0,0,0,0,0,17,0,0,12,0],[1,0,0,0,0,28,0,0,0,0,17,0,0,0,0,17],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0],[0,28,0,0,1,0,0,0,0,0,28,0,0,0,0,28] >;
 

175 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4Q7A···7F14A···14R14S···14AP28A···28BT28BU···28CX
order122222224···44···47···714···1414···1428···2828···28
size111144442···24···41···11···14···42···24···4

175 irreducible representations

dim1111111111112244
type+++++++
imageC1C2C2C2C2C2C7C14C14C14C14C14C4○D4C7×C4○D42+ (1+4)C7×2+ (1+4)
kernelC7×C22.49C24C7×C42⋊C2D4×C28C7×C4⋊D4C7×C4.4D4C7×C4⋊Q8C22.49C24C42⋊C2C4×D4C4⋊D4C4.4D4C4⋊Q8C28C4C14C2
# reps142441624122424684816

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{49}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.49C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1338);
 
// by ID
 
G=gap.SmallGroup(448,1338);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,604,1690,416]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=c,f^2=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=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀