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## G = C7×C22.34C24order 448 = 26·7

### Direct product of C7 and C22.34C24

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

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

Generators and relations for C7×C22.34C24
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 >

Subgroups: 402 in 240 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C28, C28, C2×C14, C2×C14, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C41D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22.34C24, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, C7×C42⋊C2, D4×C28, C7×C4⋊D4, C7×C22.D4, C7×C42.C2, C7×C41D4, C7×C22.34C24
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, 2+ 1+4, C22×C14, C22.34C24, C7×C4○D4, C23×C14, C14×C4○D4, C7×2+ 1+4, C7×C22.34C24

Smallest permutation representation of C7×C22.34C24
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 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 96)(86 97)(87 98)(88 92)(89 93)(90 94)(91 95)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 124)(114 125)(115 126)(116 120)(117 121)(118 122)(119 123)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 152)(142 153)(143 154)(144 148)(145 149)(146 150)(147 151)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 180)(170 181)(171 182)(172 176)(173 177)(174 178)(175 179)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 208)(198 209)(199 210)(200 204)(201 205)(202 206)(203 207)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(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)(29 41)(30 42)(31 36)(32 37)(33 38)(34 39)(35 40)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)(85 103)(86 104)(87 105)(88 99)(89 100)(90 101)(91 102)(106 123)(107 124)(108 125)(109 126)(110 120)(111 121)(112 122)(113 131)(114 132)(115 133)(116 127)(117 128)(118 129)(119 130)(134 151)(135 152)(136 153)(137 154)(138 148)(139 149)(140 150)(141 159)(142 160)(143 161)(144 155)(145 156)(146 157)(147 158)(162 179)(163 180)(164 181)(165 182)(166 176)(167 177)(168 178)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 207)(191 208)(192 209)(193 210)(194 204)(195 205)(196 206)(197 215)(198 216)(199 217)(200 211)(201 212)(202 213)(203 214)
(1 214)(2 215)(3 216)(4 217)(5 211)(6 212)(7 213)(8 54)(9 55)(10 56)(11 50)(12 51)(13 52)(14 53)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 64)(22 61)(23 62)(24 63)(25 57)(26 58)(27 59)(28 60)(29 192)(30 193)(31 194)(32 195)(33 196)(34 190)(35 191)(36 204)(37 205)(38 206)(39 207)(40 208)(41 209)(42 210)(43 200)(44 201)(45 202)(46 203)(47 197)(48 198)(49 199)(71 222)(72 223)(73 224)(74 218)(75 219)(76 220)(77 221)(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)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(99 148)(100 149)(101 150)(102 151)(103 152)(104 153)(105 154)(106 179)(107 180)(108 181)(109 182)(110 176)(111 177)(112 178)(113 187)(114 188)(115 189)(116 183)(117 184)(118 185)(119 186)(120 166)(121 167)(122 168)(123 162)(124 163)(125 164)(126 165)(127 172)(128 173)(129 174)(130 175)(131 169)(132 170)(133 171)
(1 147 46 158)(2 141 47 159)(3 142 48 160)(4 143 49 161)(5 144 43 155)(6 145 44 156)(7 146 45 157)(8 127 21 116)(9 128 15 117)(10 129 16 118)(11 130 17 119)(12 131 18 113)(13 132 19 114)(14 133 20 115)(22 121 223 111)(23 122 224 112)(24 123 218 106)(25 124 219 107)(26 125 220 108)(27 126 221 109)(28 120 222 110)(29 153 41 136)(30 154 42 137)(31 148 36 138)(32 149 37 139)(33 150 38 140)(34 151 39 134)(35 152 40 135)(50 179 67 162)(51 180 68 163)(52 181 69 164)(53 182 70 165)(54 176 64 166)(55 177 65 167)(56 178 66 168)(57 187 75 169)(58 188 76 170)(59 189 77 171)(60 183 71 172)(61 184 72 173)(62 185 73 174)(63 186 74 175)(78 207 95 190)(79 208 96 191)(80 209 97 192)(81 210 98 193)(82 204 92 194)(83 205 93 195)(84 206 94 196)(85 215 103 197)(86 216 104 198)(87 217 105 199)(88 211 99 200)(89 212 100 201)(90 213 101 202)(91 214 102 203)
(1 102)(2 103)(3 104)(4 105)(5 99)(6 100)(7 101)(8 166)(9 167)(10 168)(11 162)(12 163)(13 164)(14 165)(15 177)(16 178)(17 179)(18 180)(19 181)(20 182)(21 176)(22 173)(23 174)(24 175)(25 169)(26 170)(27 171)(28 172)(29 80)(30 81)(31 82)(32 83)(33 84)(34 78)(35 79)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 88)(44 89)(45 90)(46 91)(47 85)(48 86)(49 87)(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)(134 214)(135 215)(136 216)(137 217)(138 211)(139 212)(140 213)(141 208)(142 209)(143 210)(144 204)(145 205)(146 206)(147 207)(148 200)(149 201)(150 202)(151 203)(152 197)(153 198)(154 199)(155 194)(156 195)(157 196)(158 190)(159 191)(160 192)(161 193)(183 222)(184 223)(185 224)(186 218)(187 219)(188 220)(189 221)
(1 50 34 74)(2 51 35 75)(3 52 29 76)(4 53 30 77)(5 54 31 71)(6 55 32 72)(7 56 33 73)(8 211 222 194)(9 212 223 195)(10 213 224 196)(11 214 218 190)(12 215 219 191)(13 216 220 192)(14 217 221 193)(15 201 22 205)(16 202 23 206)(17 203 24 207)(18 197 25 208)(19 198 26 209)(20 199 27 210)(21 200 28 204)(36 60 43 64)(37 61 44 65)(38 62 45 66)(39 63 46 67)(40 57 47 68)(41 58 48 69)(42 59 49 70)(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)(85 124 96 113)(86 125 97 114)(87 126 98 115)(88 120 92 116)(89 121 93 117)(90 122 94 118)(91 123 95 119)(134 186 158 162)(135 187 159 163)(136 188 160 164)(137 189 161 165)(138 183 155 166)(139 184 156 167)(140 185 157 168)(141 180 152 169)(142 181 153 170)(143 182 154 171)(144 176 148 172)(145 177 149 173)(146 178 150 174)(147 179 151 175)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2H 4A ··· 4F 4G ··· 4M 7A ··· 7F 14A ··· 14R 14S ··· 14AV 28A ··· 28AJ 28AK ··· 28BZ order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 ··· 4 2 ··· 2 4 ··· 4 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 C4○D4 C7×C4○D4 2+ 1+4 C7×2+ 1+4 kernel C7×C22.34C24 C7×C42⋊C2 D4×C28 C7×C4⋊D4 C7×C22.D4 C7×C42.C2 C7×C4⋊1D4 C22.34C24 C42⋊C2 C4×D4 C4⋊D4 C22.D4 C42.C2 C4⋊1D4 C28 C4 C14 C2 # reps 1 1 2 6 4 1 1 6 6 12 36 24 6 6 4 24 2 12

Matrix representation of C7×C22.34C24 in GL6(𝔽29)

 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 24 0 0 0 0 27 7 0 0 0 0 0 0 16 0 3 0 0 0 26 5 1 16 0 0 2 0 13 0 0 0 2 13 23 24
,
 17 0 0 0 0 0 0 17 0 0 0 0 0 0 4 0 19 0 0 0 6 16 16 5 0 0 22 0 25 0 0 0 10 24 20 13
,
 3 27 0 0 0 0 4 26 0 0 0 0 0 0 15 0 27 0 0 0 7 0 9 1 0 0 25 0 14 0 0 0 18 1 4 0
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 20 27 0 0 0 0 12 9 0 0 0 0 7 14 0 1 0 0 11 4 28 0

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] >;

C7×C22.34C24 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

׿
×
𝔽