direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊C4⋊D7, C4⋊C4⋊41D14, (C2×C14).56C24, C4⋊Dic7⋊54C22, C14⋊3(C42⋊2C2), (C2×C28).617C23, Dic7⋊C4⋊69C22, D14⋊C4.95C22, (C4×Dic7)⋊76C22, (C22×C4).181D14, C22.90(C23×D7), C22.78(C4○D28), (C2×Dic7).17C23, (C23×D7).34C22, (C22×D7).14C23, C23.331(C22×D7), C22.75(D4⋊2D7), (C22×C28).360C22, (C22×C14).405C23, C22.37(Q8⋊2D7), (C22×Dic7).83C22, (C2×C4⋊C4)⋊21D7, (C14×C4⋊C4)⋊18C2, C7⋊3(C2×C42⋊2C2), (C2×C4×Dic7)⋊33C2, (C7×C4⋊C4)⋊49C22, (C2×C4⋊Dic7)⋊22C2, C2.25(C2×C4○D28), C14.23(C2×C4○D4), C2.8(C2×Q8⋊2D7), (C2×Dic7⋊C4)⋊45C2, (C2×D14⋊C4).26C2, C2.16(C2×D4⋊2D7), (C2×C4).144(C22×D7), (C2×C14).108(C4○D4), SmallGroup(448,965)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊C4⋊D7
G = < a,b,c,d,e | a2=b4=c4=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=bc2, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 1092 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊2C2, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C42⋊2C2, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, C23×D7, C4⋊C4⋊D7, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×D14⋊C4, C14×C4⋊C4, C2×C4⋊C4⋊D7
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C42⋊2C2, C2×C4○D4, C22×D7, C2×C42⋊2C2, C4○D28, D4⋊2D7, Q8⋊2D7, C23×D7, C4⋊C4⋊D7, C2×C4○D28, C2×D4⋊2D7, C2×Q8⋊2D7, C2×C4⋊C4⋊D7
►On 224 points
Generators in S224
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 141)(37 142)(38 143)(39 144)(40 145)(41 146)(42 147)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 176)(58 177)(59 178)(60 179)(61 180)(62 181)(63 182)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 218)(100 219)(101 220)(102 221)(103 222)(104 223)(105 224)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)
(1 162 8 155)(2 163 9 156)(3 164 10 157)(4 165 11 158)(5 166 12 159)(6 167 13 160)(7 168 14 161)(15 148 22 141)(16 149 23 142)(17 150 24 143)(18 151 25 144)(19 152 26 145)(20 153 27 146)(21 154 28 147)(29 127 36 134)(30 128 37 135)(31 129 38 136)(32 130 39 137)(33 131 40 138)(34 132 41 139)(35 133 42 140)(43 113 50 120)(44 114 51 121)(45 115 52 122)(46 116 53 123)(47 117 54 124)(48 118 55 125)(49 119 56 126)(57 211 64 218)(58 212 65 219)(59 213 66 220)(60 214 67 221)(61 215 68 222)(62 216 69 223)(63 217 70 224)(71 197 78 204)(72 198 79 205)(73 199 80 206)(74 200 81 207)(75 201 82 208)(76 202 83 209)(77 203 84 210)(85 190 92 183)(86 191 93 184)(87 192 94 185)(88 193 95 186)(89 194 96 187)(90 195 97 188)(91 196 98 189)(99 176 106 169)(100 177 107 170)(101 178 108 171)(102 179 109 172)(103 180 110 173)(104 181 111 174)(105 182 112 175)
(1 71 15 57)(2 72 16 58)(3 73 17 59)(4 74 18 60)(5 75 19 61)(6 76 20 62)(7 77 21 63)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)(113 183 127 169)(114 184 128 170)(115 185 129 171)(116 186 130 172)(117 187 131 173)(118 188 132 174)(119 189 133 175)(120 190 134 176)(121 191 135 177)(122 192 136 178)(123 193 137 179)(124 194 138 180)(125 195 139 181)(126 196 140 182)(141 211 155 197)(142 212 156 198)(143 213 157 199)(144 214 158 200)(145 215 159 201)(146 216 160 202)(147 217 161 203)(148 218 162 204)(149 219 163 205)(150 220 164 206)(151 221 165 207)(152 222 166 208)(153 223 167 209)(154 224 168 210)
(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 7)(2 6)(3 5)(8 14)(9 13)(10 12)(15 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(113 119)(114 118)(115 117)(120 126)(121 125)(122 124)(127 133)(128 132)(129 131)(134 140)(135 139)(136 138)(141 161)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(197 224)(198 223)(199 222)(200 221)(201 220)(202 219)(203 218)(204 217)(205 216)(206 215)(207 214)(208 213)(209 212)(210 211)
G:=sub<Sym(224)| (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,162)(44,163)(45,164)(46,165)(47,166)(48,167)(49,168)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,162,8,155)(2,163,9,156)(3,164,10,157)(4,165,11,158)(5,166,12,159)(6,167,13,160)(7,168,14,161)(15,148,22,141)(16,149,23,142)(17,150,24,143)(18,151,25,144)(19,152,26,145)(20,153,27,146)(21,154,28,147)(29,127,36,134)(30,128,37,135)(31,129,38,136)(32,130,39,137)(33,131,40,138)(34,132,41,139)(35,133,42,140)(43,113,50,120)(44,114,51,121)(45,115,52,122)(46,116,53,123)(47,117,54,124)(48,118,55,125)(49,119,56,126)(57,211,64,218)(58,212,65,219)(59,213,66,220)(60,214,67,221)(61,215,68,222)(62,216,69,223)(63,217,70,224)(71,197,78,204)(72,198,79,205)(73,199,80,206)(74,200,81,207)(75,201,82,208)(76,202,83,209)(77,203,84,210)(85,190,92,183)(86,191,93,184)(87,192,94,185)(88,193,95,186)(89,194,96,187)(90,195,97,188)(91,196,98,189)(99,176,106,169)(100,177,107,170)(101,178,108,171)(102,179,109,172)(103,180,110,173)(104,181,111,174)(105,182,112,175), (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,183,127,169)(114,184,128,170)(115,185,129,171)(116,186,130,172)(117,187,131,173)(118,188,132,174)(119,189,133,175)(120,190,134,176)(121,191,135,177)(122,192,136,178)(123,193,137,179)(124,194,138,180)(125,195,139,181)(126,196,140,182)(141,211,155,197)(142,212,156,198)(143,213,157,199)(144,214,158,200)(145,215,159,201)(146,216,160,202)(147,217,161,203)(148,218,162,204)(149,219,163,205)(150,220,164,206)(151,221,165,207)(152,222,166,208)(153,223,167,209)(154,224,168,210), (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), (1,7)(2,6)(3,5)(8,14)(9,13)(10,12)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(113,119)(114,118)(115,117)(120,126)(121,125)(122,124)(127,133)(128,132)(129,131)(134,140)(135,139)(136,138)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,224)(198,223)(199,222)(200,221)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211)>;
G:=Group( (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,162)(44,163)(45,164)(46,165)(47,166)(48,167)(49,168)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,162,8,155)(2,163,9,156)(3,164,10,157)(4,165,11,158)(5,166,12,159)(6,167,13,160)(7,168,14,161)(15,148,22,141)(16,149,23,142)(17,150,24,143)(18,151,25,144)(19,152,26,145)(20,153,27,146)(21,154,28,147)(29,127,36,134)(30,128,37,135)(31,129,38,136)(32,130,39,137)(33,131,40,138)(34,132,41,139)(35,133,42,140)(43,113,50,120)(44,114,51,121)(45,115,52,122)(46,116,53,123)(47,117,54,124)(48,118,55,125)(49,119,56,126)(57,211,64,218)(58,212,65,219)(59,213,66,220)(60,214,67,221)(61,215,68,222)(62,216,69,223)(63,217,70,224)(71,197,78,204)(72,198,79,205)(73,199,80,206)(74,200,81,207)(75,201,82,208)(76,202,83,209)(77,203,84,210)(85,190,92,183)(86,191,93,184)(87,192,94,185)(88,193,95,186)(89,194,96,187)(90,195,97,188)(91,196,98,189)(99,176,106,169)(100,177,107,170)(101,178,108,171)(102,179,109,172)(103,180,110,173)(104,181,111,174)(105,182,112,175), (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,183,127,169)(114,184,128,170)(115,185,129,171)(116,186,130,172)(117,187,131,173)(118,188,132,174)(119,189,133,175)(120,190,134,176)(121,191,135,177)(122,192,136,178)(123,193,137,179)(124,194,138,180)(125,195,139,181)(126,196,140,182)(141,211,155,197)(142,212,156,198)(143,213,157,199)(144,214,158,200)(145,215,159,201)(146,216,160,202)(147,217,161,203)(148,218,162,204)(149,219,163,205)(150,220,164,206)(151,221,165,207)(152,222,166,208)(153,223,167,209)(154,224,168,210), (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), (1,7)(2,6)(3,5)(8,14)(9,13)(10,12)(15,21)(16,20)(17,19)(22,28)(23,27)(24,26)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(113,119)(114,118)(115,117)(120,126)(121,125)(122,124)(127,133)(128,132)(129,131)(134,140)(135,139)(136,138)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,224)(198,223)(199,222)(200,221)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)(207,214)(208,213)(209,212)(210,211) );
G=PermutationGroup([[(1,120),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,148),(30,149),(31,150),(32,151),(33,152),(34,153),(35,154),(36,141),(37,142),(38,143),(39,144),(40,145),(41,146),(42,147),(43,162),(44,163),(45,164),(46,165),(47,166),(48,167),(49,168),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,161),(57,176),(58,177),(59,178),(60,179),(61,180),(62,181),(63,182),(64,169),(65,170),(66,171),(67,172),(68,173),(69,174),(70,175),(71,190),(72,191),(73,192),(74,193),(75,194),(76,195),(77,196),(78,183),(79,184),(80,185),(81,186),(82,187),(83,188),(84,189),(85,204),(86,205),(87,206),(88,207),(89,208),(90,209),(91,210),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,218),(100,219),(101,220),(102,221),(103,222),(104,223),(105,224),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217)], [(1,162,8,155),(2,163,9,156),(3,164,10,157),(4,165,11,158),(5,166,12,159),(6,167,13,160),(7,168,14,161),(15,148,22,141),(16,149,23,142),(17,150,24,143),(18,151,25,144),(19,152,26,145),(20,153,27,146),(21,154,28,147),(29,127,36,134),(30,128,37,135),(31,129,38,136),(32,130,39,137),(33,131,40,138),(34,132,41,139),(35,133,42,140),(43,113,50,120),(44,114,51,121),(45,115,52,122),(46,116,53,123),(47,117,54,124),(48,118,55,125),(49,119,56,126),(57,211,64,218),(58,212,65,219),(59,213,66,220),(60,214,67,221),(61,215,68,222),(62,216,69,223),(63,217,70,224),(71,197,78,204),(72,198,79,205),(73,199,80,206),(74,200,81,207),(75,201,82,208),(76,202,83,209),(77,203,84,210),(85,190,92,183),(86,191,93,184),(87,192,94,185),(88,193,95,186),(89,194,96,187),(90,195,97,188),(91,196,98,189),(99,176,106,169),(100,177,107,170),(101,178,108,171),(102,179,109,172),(103,180,110,173),(104,181,111,174),(105,182,112,175)], [(1,71,15,57),(2,72,16,58),(3,73,17,59),(4,74,18,60),(5,75,19,61),(6,76,20,62),(7,77,21,63),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,99,43,85),(30,100,44,86),(31,101,45,87),(32,102,46,88),(33,103,47,89),(34,104,48,90),(35,105,49,91),(36,106,50,92),(37,107,51,93),(38,108,52,94),(39,109,53,95),(40,110,54,96),(41,111,55,97),(42,112,56,98),(113,183,127,169),(114,184,128,170),(115,185,129,171),(116,186,130,172),(117,187,131,173),(118,188,132,174),(119,189,133,175),(120,190,134,176),(121,191,135,177),(122,192,136,178),(123,193,137,179),(124,194,138,180),(125,195,139,181),(126,196,140,182),(141,211,155,197),(142,212,156,198),(143,213,157,199),(144,214,158,200),(145,215,159,201),(146,216,160,202),(147,217,161,203),(148,218,162,204),(149,219,163,205),(150,220,164,206),(151,221,165,207),(152,222,166,208),(153,223,167,209),(154,224,168,210)], [(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)], [(1,7),(2,6),(3,5),(8,14),(9,13),(10,12),(15,21),(16,20),(17,19),(22,28),(23,27),(24,26),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(85,112),(86,111),(87,110),(88,109),(89,108),(90,107),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99),(113,119),(114,118),(115,117),(120,126),(121,125),(122,124),(127,133),(128,132),(129,131),(134,140),(135,139),(136,138),(141,161),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,168),(149,167),(150,166),(151,165),(152,164),(153,163),(154,162),(169,182),(170,181),(171,180),(172,179),(173,178),(174,177),(175,176),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(197,224),(198,223),(199,222),(200,221),(201,220),(202,219),(203,218),(204,217),(205,216),(206,215),(207,214),(208,213),(209,212),(210,211)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | C4○D28 | D4⋊2D7 | Q8⋊2D7 |
| kernel | C2×C4⋊C4⋊D7 | C4⋊C4⋊D7 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×D14⋊C4 | C14×C4⋊C4 | C2×C4⋊C4 | C2×C14 | C4⋊C4 | C22×C4 | C22 | C22 | C22 |
| # reps | 1 | 8 | 1 | 1 | 1 | 3 | 1 | 3 | 12 | 12 | 9 | 24 | 6 | 6 |
Matrix representation of C2×C4⋊C4⋊D7 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 16 | 0 | 0 |
| 0 | 26 | 22 | 0 | 0 |
| 0 | 0 | 0 | 12 | 18 |
| 0 | 0 | 0 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 17 | 11 |
| 0 | 0 | 0 | 16 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 21 | 28 | 0 | 0 |
| 0 | 2 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 20 | 21 | 0 | 0 |
| 0 | 10 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,7,26,0,0,0,16,22,0,0,0,0,0,12,0,0,0,0,18,17],[1,0,0,0,0,0,17,0,0,0,0,0,17,0,0,0,0,0,17,16,0,0,0,11,12],[1,0,0,0,0,0,21,2,0,0,0,28,11,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,20,10,0,0,0,21,9,0,0,0,0,0,1,0,0,0,0,16,28] >;
C2×C4⋊C4⋊D7 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes C_4\rtimes D_7
% in TeX
G:=Group("C2xC4:C4:D7"); // GroupNames label
G:=SmallGroup(448,965);
// by ID
G=gap.SmallGroup(448,965);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,100,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b*c^2,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations