direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C42⋊D7, C42⋊35D14, (C2×C42)⋊3D7, (C4×C28)⋊55C22, C14.3(C23×C4), (C2×C14).16C24, C14⋊1(C42⋊C2), (C2×C28).693C23, C28.141(C22×C4), Dic7⋊C4⋊77C22, (C4×Dic7)⋊70C22, D14.17(C22×C4), (C22×C4).467D14, C22.13(C23×D7), D14⋊C4.161C22, C22.67(C4○D28), Dic7.17(C22×C4), (C23×D7).91C22, C23.313(C22×D7), (C22×C28).502C22, (C22×C14).378C23, (C2×Dic7).172C23, (C22×D7).145C23, (C22×Dic7).200C22, (C2×C4×C28)⋊17C2, (C2×C4×D7)⋊13C4, C4.116(C2×C4×D7), (C4×D7)⋊13(C2×C4), C2.5(D7×C22×C4), C7⋊1(C2×C42⋊C2), (C2×C4×Dic7)⋊27C2, C2.2(C2×C4○D28), C14.4(C2×C4○D4), C22.68(C2×C4×D7), (C2×C4).185(C4×D7), (D7×C22×C4).21C2, (C2×C28).255(C2×C4), (C2×Dic7⋊C4)⋊51C2, (C2×D14⋊C4).29C2, (C2×C4×D7).286C22, (C2×C14).95(C4○D4), (C22×D7).64(C2×C4), (C2×C4).648(C22×D7), (C2×C14).146(C22×C4), (C2×Dic7).103(C2×C4), SmallGroup(448,925)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C42⋊D7
G = < a,b,c,d,e | a2=b4=c4=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 1284 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C42⋊C2, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C2×C4×D7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C23×D7, C42⋊D7, C2×C4×Dic7, C2×Dic7⋊C4, C2×D14⋊C4, C2×C4×C28, D7×C22×C4, C2×C42⋊D7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, C24, D14, C42⋊C2, C23×C4, C2×C4○D4, C4×D7, C22×D7, C2×C42⋊C2, C2×C4×D7, C4○D28, C23×D7, C42⋊D7, D7×C22×C4, C2×C4○D28, C2×C42⋊D7
►On 224 points
Generators in S224
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 162)(30 163)(31 164)(32 165)(33 166)(34 167)(35 168)(36 155)(37 156)(38 157)(39 158)(40 159)(41 160)(42 161)(43 148)(44 149)(45 150)(46 151)(47 152)(48 153)(49 154)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 190)(58 191)(59 192)(60 193)(61 194)(62 195)(63 196)(64 183)(65 184)(66 185)(67 186)(68 187)(69 188)(70 189)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 169)(79 170)(80 171)(81 172)(82 173)(83 174)(84 175)(85 218)(86 219)(87 220)(88 221)(89 222)(90 223)(91 224)(92 211)(93 212)(94 213)(95 214)(96 215)(97 216)(98 217)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 197)(107 198)(108 199)(109 200)(110 201)(111 202)(112 203)
(1 190 8 183)(2 191 9 184)(3 192 10 185)(4 193 11 186)(5 194 12 187)(6 195 13 188)(7 196 14 189)(15 176 22 169)(16 177 23 170)(17 178 24 171)(18 179 25 172)(19 180 26 173)(20 181 27 174)(21 182 28 175)(29 218 36 211)(30 219 37 212)(31 220 38 213)(32 221 39 214)(33 222 40 215)(34 223 41 216)(35 224 42 217)(43 204 50 197)(44 205 51 198)(45 206 52 199)(46 207 53 200)(47 208 54 201)(48 209 55 202)(49 210 56 203)(57 127 64 134)(58 128 65 135)(59 129 66 136)(60 130 67 137)(61 131 68 138)(62 132 69 139)(63 133 70 140)(71 113 78 120)(72 114 79 121)(73 115 80 122)(74 116 81 123)(75 117 82 124)(76 118 83 125)(77 119 84 126)(85 155 92 162)(86 156 93 163)(87 157 94 164)(88 158 95 165)(89 159 96 166)(90 160 97 167)(91 161 98 168)(99 141 106 148)(100 142 107 149)(101 143 108 150)(102 144 109 151)(103 145 110 152)(104 146 111 153)(105 147 112 154)
(1 106 22 85)(2 107 23 86)(3 108 24 87)(4 109 25 88)(5 110 26 89)(6 111 27 90)(7 112 28 91)(8 99 15 92)(9 100 16 93)(10 101 17 94)(11 102 18 95)(12 103 19 96)(13 104 20 97)(14 105 21 98)(29 64 50 71)(30 65 51 72)(31 66 52 73)(32 67 53 74)(33 68 54 75)(34 69 55 76)(35 70 56 77)(36 57 43 78)(37 58 44 79)(38 59 45 80)(39 60 46 81)(40 61 47 82)(41 62 48 83)(42 63 49 84)(113 218 134 197)(114 219 135 198)(115 220 136 199)(116 221 137 200)(117 222 138 201)(118 223 139 202)(119 224 140 203)(120 211 127 204)(121 212 128 205)(122 213 129 206)(123 214 130 207)(124 215 131 208)(125 216 132 209)(126 217 133 210)(141 176 162 183)(142 177 163 184)(143 178 164 185)(144 179 165 186)(145 180 166 187)(146 181 167 188)(147 182 168 189)(148 169 155 190)(149 170 156 191)(150 171 157 192)(151 172 158 193)(152 173 159 194)(153 174 160 195)(154 175 161 196)
(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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 140)(9 139)(10 138)(11 137)(12 136)(13 135)(14 134)(15 119)(16 118)(17 117)(18 116)(19 115)(20 114)(21 113)(22 126)(23 125)(24 124)(25 123)(26 122)(27 121)(28 120)(29 168)(30 167)(31 166)(32 165)(33 164)(34 163)(35 162)(36 161)(37 160)(38 159)(39 158)(40 157)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 189)(58 188)(59 187)(60 186)(61 185)(62 184)(63 183)(64 196)(65 195)(66 194)(67 193)(68 192)(69 191)(70 190)(71 175)(72 174)(73 173)(74 172)(75 171)(76 170)(77 169)(78 182)(79 181)(80 180)(81 179)(82 178)(83 177)(84 176)(85 224)(86 223)(87 222)(88 221)(89 220)(90 219)(91 218)(92 217)(93 216)(94 215)(95 214)(96 213)(97 212)(98 211)(99 210)(100 209)(101 208)(102 207)(103 206)(104 205)(105 204)(106 203)(107 202)(108 201)(109 200)(110 199)(111 198)(112 197)
G:=sub<Sym(224)| (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,155)(37,156)(38,157)(39,158)(40,159)(41,160)(42,161)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,211)(93,212)(94,213)(95,214)(96,215)(97,216)(98,217)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,197)(107,198)(108,199)(109,200)(110,201)(111,202)(112,203), (1,190,8,183)(2,191,9,184)(3,192,10,185)(4,193,11,186)(5,194,12,187)(6,195,13,188)(7,196,14,189)(15,176,22,169)(16,177,23,170)(17,178,24,171)(18,179,25,172)(19,180,26,173)(20,181,27,174)(21,182,28,175)(29,218,36,211)(30,219,37,212)(31,220,38,213)(32,221,39,214)(33,222,40,215)(34,223,41,216)(35,224,42,217)(43,204,50,197)(44,205,51,198)(45,206,52,199)(46,207,53,200)(47,208,54,201)(48,209,55,202)(49,210,56,203)(57,127,64,134)(58,128,65,135)(59,129,66,136)(60,130,67,137)(61,131,68,138)(62,132,69,139)(63,133,70,140)(71,113,78,120)(72,114,79,121)(73,115,80,122)(74,116,81,123)(75,117,82,124)(76,118,83,125)(77,119,84,126)(85,155,92,162)(86,156,93,163)(87,157,94,164)(88,158,95,165)(89,159,96,166)(90,160,97,167)(91,161,98,168)(99,141,106,148)(100,142,107,149)(101,143,108,150)(102,144,109,151)(103,145,110,152)(104,146,111,153)(105,147,112,154), (1,106,22,85)(2,107,23,86)(3,108,24,87)(4,109,25,88)(5,110,26,89)(6,111,27,90)(7,112,28,91)(8,99,15,92)(9,100,16,93)(10,101,17,94)(11,102,18,95)(12,103,19,96)(13,104,20,97)(14,105,21,98)(29,64,50,71)(30,65,51,72)(31,66,52,73)(32,67,53,74)(33,68,54,75)(34,69,55,76)(35,70,56,77)(36,57,43,78)(37,58,44,79)(38,59,45,80)(39,60,46,81)(40,61,47,82)(41,62,48,83)(42,63,49,84)(113,218,134,197)(114,219,135,198)(115,220,136,199)(116,221,137,200)(117,222,138,201)(118,223,139,202)(119,224,140,203)(120,211,127,204)(121,212,128,205)(122,213,129,206)(123,214,130,207)(124,215,131,208)(125,216,132,209)(126,217,133,210)(141,176,162,183)(142,177,163,184)(143,178,164,185)(144,179,165,186)(145,180,166,187)(146,181,167,188)(147,182,168,189)(148,169,155,190)(149,170,156,191)(150,171,157,192)(151,172,158,193)(152,173,159,194)(153,174,160,195)(154,175,161,196), (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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,120)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,189)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,196)(65,195)(66,194)(67,193)(68,192)(69,191)(70,190)(71,175)(72,174)(73,173)(74,172)(75,171)(76,170)(77,169)(78,182)(79,181)(80,180)(81,179)(82,178)(83,177)(84,176)(85,224)(86,223)(87,222)(88,221)(89,220)(90,219)(91,218)(92,217)(93,216)(94,215)(95,214)(96,213)(97,212)(98,211)(99,210)(100,209)(101,208)(102,207)(103,206)(104,205)(105,204)(106,203)(107,202)(108,201)(109,200)(110,199)(111,198)(112,197)>;
G:=Group( (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,155)(37,156)(38,157)(39,158)(40,159)(41,160)(42,161)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,211)(93,212)(94,213)(95,214)(96,215)(97,216)(98,217)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,197)(107,198)(108,199)(109,200)(110,201)(111,202)(112,203), (1,190,8,183)(2,191,9,184)(3,192,10,185)(4,193,11,186)(5,194,12,187)(6,195,13,188)(7,196,14,189)(15,176,22,169)(16,177,23,170)(17,178,24,171)(18,179,25,172)(19,180,26,173)(20,181,27,174)(21,182,28,175)(29,218,36,211)(30,219,37,212)(31,220,38,213)(32,221,39,214)(33,222,40,215)(34,223,41,216)(35,224,42,217)(43,204,50,197)(44,205,51,198)(45,206,52,199)(46,207,53,200)(47,208,54,201)(48,209,55,202)(49,210,56,203)(57,127,64,134)(58,128,65,135)(59,129,66,136)(60,130,67,137)(61,131,68,138)(62,132,69,139)(63,133,70,140)(71,113,78,120)(72,114,79,121)(73,115,80,122)(74,116,81,123)(75,117,82,124)(76,118,83,125)(77,119,84,126)(85,155,92,162)(86,156,93,163)(87,157,94,164)(88,158,95,165)(89,159,96,166)(90,160,97,167)(91,161,98,168)(99,141,106,148)(100,142,107,149)(101,143,108,150)(102,144,109,151)(103,145,110,152)(104,146,111,153)(105,147,112,154), (1,106,22,85)(2,107,23,86)(3,108,24,87)(4,109,25,88)(5,110,26,89)(6,111,27,90)(7,112,28,91)(8,99,15,92)(9,100,16,93)(10,101,17,94)(11,102,18,95)(12,103,19,96)(13,104,20,97)(14,105,21,98)(29,64,50,71)(30,65,51,72)(31,66,52,73)(32,67,53,74)(33,68,54,75)(34,69,55,76)(35,70,56,77)(36,57,43,78)(37,58,44,79)(38,59,45,80)(39,60,46,81)(40,61,47,82)(41,62,48,83)(42,63,49,84)(113,218,134,197)(114,219,135,198)(115,220,136,199)(116,221,137,200)(117,222,138,201)(118,223,139,202)(119,224,140,203)(120,211,127,204)(121,212,128,205)(122,213,129,206)(123,214,130,207)(124,215,131,208)(125,216,132,209)(126,217,133,210)(141,176,162,183)(142,177,163,184)(143,178,164,185)(144,179,165,186)(145,180,166,187)(146,181,167,188)(147,182,168,189)(148,169,155,190)(149,170,156,191)(150,171,157,192)(151,172,158,193)(152,173,159,194)(153,174,160,195)(154,175,161,196), (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), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,140)(9,139)(10,138)(11,137)(12,136)(13,135)(14,134)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,126)(23,125)(24,124)(25,123)(26,122)(27,121)(28,120)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,189)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,196)(65,195)(66,194)(67,193)(68,192)(69,191)(70,190)(71,175)(72,174)(73,173)(74,172)(75,171)(76,170)(77,169)(78,182)(79,181)(80,180)(81,179)(82,178)(83,177)(84,176)(85,224)(86,223)(87,222)(88,221)(89,220)(90,219)(91,218)(92,217)(93,216)(94,215)(95,214)(96,213)(97,212)(98,211)(99,210)(100,209)(101,208)(102,207)(103,206)(104,205)(105,204)(106,203)(107,202)(108,201)(109,200)(110,199)(111,198)(112,197) );
G=PermutationGroup([[(1,134),(2,135),(3,136),(4,137),(5,138),(6,139),(7,140),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,120),(16,121),(17,122),(18,123),(19,124),(20,125),(21,126),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,162),(30,163),(31,164),(32,165),(33,166),(34,167),(35,168),(36,155),(37,156),(38,157),(39,158),(40,159),(41,160),(42,161),(43,148),(44,149),(45,150),(46,151),(47,152),(48,153),(49,154),(50,141),(51,142),(52,143),(53,144),(54,145),(55,146),(56,147),(57,190),(58,191),(59,192),(60,193),(61,194),(62,195),(63,196),(64,183),(65,184),(66,185),(67,186),(68,187),(69,188),(70,189),(71,176),(72,177),(73,178),(74,179),(75,180),(76,181),(77,182),(78,169),(79,170),(80,171),(81,172),(82,173),(83,174),(84,175),(85,218),(86,219),(87,220),(88,221),(89,222),(90,223),(91,224),(92,211),(93,212),(94,213),(95,214),(96,215),(97,216),(98,217),(99,204),(100,205),(101,206),(102,207),(103,208),(104,209),(105,210),(106,197),(107,198),(108,199),(109,200),(110,201),(111,202),(112,203)], [(1,190,8,183),(2,191,9,184),(3,192,10,185),(4,193,11,186),(5,194,12,187),(6,195,13,188),(7,196,14,189),(15,176,22,169),(16,177,23,170),(17,178,24,171),(18,179,25,172),(19,180,26,173),(20,181,27,174),(21,182,28,175),(29,218,36,211),(30,219,37,212),(31,220,38,213),(32,221,39,214),(33,222,40,215),(34,223,41,216),(35,224,42,217),(43,204,50,197),(44,205,51,198),(45,206,52,199),(46,207,53,200),(47,208,54,201),(48,209,55,202),(49,210,56,203),(57,127,64,134),(58,128,65,135),(59,129,66,136),(60,130,67,137),(61,131,68,138),(62,132,69,139),(63,133,70,140),(71,113,78,120),(72,114,79,121),(73,115,80,122),(74,116,81,123),(75,117,82,124),(76,118,83,125),(77,119,84,126),(85,155,92,162),(86,156,93,163),(87,157,94,164),(88,158,95,165),(89,159,96,166),(90,160,97,167),(91,161,98,168),(99,141,106,148),(100,142,107,149),(101,143,108,150),(102,144,109,151),(103,145,110,152),(104,146,111,153),(105,147,112,154)], [(1,106,22,85),(2,107,23,86),(3,108,24,87),(4,109,25,88),(5,110,26,89),(6,111,27,90),(7,112,28,91),(8,99,15,92),(9,100,16,93),(10,101,17,94),(11,102,18,95),(12,103,19,96),(13,104,20,97),(14,105,21,98),(29,64,50,71),(30,65,51,72),(31,66,52,73),(32,67,53,74),(33,68,54,75),(34,69,55,76),(35,70,56,77),(36,57,43,78),(37,58,44,79),(38,59,45,80),(39,60,46,81),(40,61,47,82),(41,62,48,83),(42,63,49,84),(113,218,134,197),(114,219,135,198),(115,220,136,199),(116,221,137,200),(117,222,138,201),(118,223,139,202),(119,224,140,203),(120,211,127,204),(121,212,128,205),(122,213,129,206),(123,214,130,207),(124,215,131,208),(125,216,132,209),(126,217,133,210),(141,176,162,183),(142,177,163,184),(143,178,164,185),(144,179,165,186),(145,180,166,187),(146,181,167,188),(147,182,168,189),(148,169,155,190),(149,170,156,191),(150,171,157,192),(151,172,158,193),(152,173,159,194),(153,174,160,195),(154,175,161,196)], [(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)], [(1,133),(2,132),(3,131),(4,130),(5,129),(6,128),(7,127),(8,140),(9,139),(10,138),(11,137),(12,136),(13,135),(14,134),(15,119),(16,118),(17,117),(18,116),(19,115),(20,114),(21,113),(22,126),(23,125),(24,124),(25,123),(26,122),(27,121),(28,120),(29,168),(30,167),(31,166),(32,165),(33,164),(34,163),(35,162),(36,161),(37,160),(38,159),(39,158),(40,157),(41,156),(42,155),(43,154),(44,153),(45,152),(46,151),(47,150),(48,149),(49,148),(50,147),(51,146),(52,145),(53,144),(54,143),(55,142),(56,141),(57,189),(58,188),(59,187),(60,186),(61,185),(62,184),(63,183),(64,196),(65,195),(66,194),(67,193),(68,192),(69,191),(70,190),(71,175),(72,174),(73,173),(74,172),(75,171),(76,170),(77,169),(78,182),(79,181),(80,180),(81,179),(82,178),(83,177),(84,176),(85,224),(86,223),(87,222),(88,221),(89,220),(90,219),(91,218),(92,217),(93,216),(94,215),(95,214),(96,213),(97,212),(98,211),(99,210),(100,209),(101,208),(102,207),(103,206),(104,205),(105,204),(106,203),(107,202),(108,201),(109,200),(110,199),(111,198),(112,197)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 1 | ··· | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | D14 | D14 | C4×D7 | C4○D28 |
| kernel | C2×C42⋊D7 | C42⋊D7 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×D14⋊C4 | C2×C4×C28 | D7×C22×C4 | C2×C4×D7 | C2×C42 | C2×C14 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 1 | 2 | 2 | 1 | 1 | 16 | 3 | 8 | 12 | 9 | 24 | 48 |
Matrix representation of C2×C42⋊D7 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 17 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 11 | 13 |
| 0 | 0 | 22 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 28 |
| 0 | 0 | 5 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 11 | 26 |
| 0 | 0 | 11 | 18 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,17,0,0,0,0,17],[17,0,0,0,0,12,0,0,0,0,11,22,0,0,13,18],[1,0,0,0,0,1,0,0,0,0,4,5,0,0,28,28],[1,0,0,0,0,28,0,0,0,0,11,11,0,0,26,18] >;
C2×C42⋊D7 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes D_7
% in TeX
G:=Group("C2xC4^2:D7"); // GroupNames label
G:=SmallGroup(448,925);
// by ID
G=gap.SmallGroup(448,925);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,1123,80,18822]);
// 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,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations