direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D7×C4⋊C4, C28⋊2(C22×C4), D14.57(C2×D4), D14.13(C2×Q8), C22.32(Q8×D7), C14.11(C23×C4), (C2×C14).45C24, C4⋊Dic7⋊69C22, Dic7⋊4(C22×C4), C14.40(C22×D4), C22.130(D4×D7), (C22×D7).12Q8, C14.23(C22×Q8), (C2×C28).577C23, Dic7⋊C4⋊60C22, (C22×D7).109D4, D14.22(C22×C4), (C22×C4).316D14, C22.21(C23×D7), C23.324(C22×D7), (C22×C28).357C22, (C22×C14).394C23, (C2×Dic7).184C23, (C23×D7).122C22, (C22×D7).253C23, (C22×Dic7).209C22, C4⋊4(C2×C4×D7), (C2×C4×D7)⋊5C4, C14⋊1(C2×C4⋊C4), C2.3(C2×D4×D7), C2.2(C2×Q8×D7), C7⋊1(C22×C4⋊C4), (C14×C4⋊C4)⋊7C2, (C2×C28)⋊7(C2×C4), (C2×C4)⋊15(C4×D7), (C4×D7)⋊11(C2×C4), (D7×C22×C4).3C2, (C7×C4⋊C4)⋊42C22, C22.71(C2×C4×D7), C2.13(D7×C22×C4), (C2×C4⋊Dic7)⋊37C2, (C2×C14).92(C2×Q8), (C2×Dic7⋊C4)⋊37C2, (C2×Dic7)⋊22(C2×C4), (C2×C14).386(C2×D4), (C2×C4×D7).242C22, (C22×D7).74(C2×C4), (C2×C4).264(C22×D7), (C2×C14).150(C22×C4), SmallGroup(448,954)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D7×C4⋊C4
G = < a,b,c,d,e | a2=b7=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 1668 in 418 conjugacy classes, 207 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×C14, C22×C4⋊C4, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C23×D7, D7×C4⋊C4, C2×Dic7⋊C4, C2×C4⋊Dic7, C14×C4⋊C4, D7×C22×C4, D7×C22×C4, C2×D7×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, D14, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C4×D7, C22×D7, C22×C4⋊C4, C2×C4×D7, D4×D7, Q8×D7, C23×D7, D7×C4⋊C4, D7×C22×C4, C2×D4×D7, C2×Q8×D7, C2×D7×C4⋊C4
►On 224 points
Generators in S224
(1 125)(2 126)(3 120)(4 121)(5 122)(6 123)(7 124)(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 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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)(113 122)(114 121)(115 120)(116 126)(117 125)(118 124)(119 123)(127 136)(128 135)(129 134)(130 140)(131 139)(132 138)(133 137)(141 150)(142 149)(143 148)(144 154)(145 153)(146 152)(147 151)(155 164)(156 163)(157 162)(158 168)(159 167)(160 166)(161 165)(169 178)(170 177)(171 176)(172 182)(173 181)(174 180)(175 179)(183 192)(184 191)(185 190)(186 196)(187 195)(188 194)(189 193)(197 206)(198 205)(199 204)(200 210)(201 209)(202 208)(203 207)(211 220)(212 219)(213 218)(214 224)(215 223)(216 222)(217 221)
(1 195 13 188)(2 196 14 189)(3 190 8 183)(4 191 9 184)(5 192 10 185)(6 193 11 186)(7 194 12 187)(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 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 106 71 92)(58 107 72 93)(59 108 73 94)(60 109 74 95)(61 110 75 96)(62 111 76 97)(63 112 77 98)(64 99 78 85)(65 100 79 86)(66 101 80 87)(67 102 81 88)(68 103 82 89)(69 104 83 90)(70 105 84 91)(113 155 127 141)(114 156 128 142)(115 157 129 143)(116 158 130 144)(117 159 131 145)(118 160 132 146)(119 161 133 147)(120 162 134 148)(121 163 135 149)(122 164 136 150)(123 165 137 151)(124 166 138 152)(125 167 139 153)(126 168 140 154)(169 218 183 204)(170 219 184 205)(171 220 185 206)(172 221 186 207)(173 222 187 208)(174 223 188 209)(175 224 189 210)(176 211 190 197)(177 212 191 198)(178 213 192 199)(179 214 193 200)(180 215 194 201)(181 216 195 202)(182 217 196 203)
G:=sub<Sym(224)| (1,125)(2,126)(3,120)(4,121)(5,122)(6,123)(7,124)(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), (1,2,3,4,5,6,7)(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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)(113,122)(114,121)(115,120)(116,126)(117,125)(118,124)(119,123)(127,136)(128,135)(129,134)(130,140)(131,139)(132,138)(133,137)(141,150)(142,149)(143,148)(144,154)(145,153)(146,152)(147,151)(155,164)(156,163)(157,162)(158,168)(159,167)(160,166)(161,165)(169,178)(170,177)(171,176)(172,182)(173,181)(174,180)(175,179)(183,192)(184,191)(185,190)(186,196)(187,195)(188,194)(189,193)(197,206)(198,205)(199,204)(200,210)(201,209)(202,208)(203,207)(211,220)(212,219)(213,218)(214,224)(215,223)(216,222)(217,221), (1,195,13,188)(2,196,14,189)(3,190,8,183)(4,191,9,184)(5,192,10,185)(6,193,11,186)(7,194,12,187)(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), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91)(113,155,127,141)(114,156,128,142)(115,157,129,143)(116,158,130,144)(117,159,131,145)(118,160,132,146)(119,161,133,147)(120,162,134,148)(121,163,135,149)(122,164,136,150)(123,165,137,151)(124,166,138,152)(125,167,139,153)(126,168,140,154)(169,218,183,204)(170,219,184,205)(171,220,185,206)(172,221,186,207)(173,222,187,208)(174,223,188,209)(175,224,189,210)(176,211,190,197)(177,212,191,198)(178,213,192,199)(179,214,193,200)(180,215,194,201)(181,216,195,202)(182,217,196,203)>;
G:=Group( (1,125)(2,126)(3,120)(4,121)(5,122)(6,123)(7,124)(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), (1,2,3,4,5,6,7)(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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,14)(7,13)(15,24)(16,23)(17,22)(18,28)(19,27)(20,26)(21,25)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)(113,122)(114,121)(115,120)(116,126)(117,125)(118,124)(119,123)(127,136)(128,135)(129,134)(130,140)(131,139)(132,138)(133,137)(141,150)(142,149)(143,148)(144,154)(145,153)(146,152)(147,151)(155,164)(156,163)(157,162)(158,168)(159,167)(160,166)(161,165)(169,178)(170,177)(171,176)(172,182)(173,181)(174,180)(175,179)(183,192)(184,191)(185,190)(186,196)(187,195)(188,194)(189,193)(197,206)(198,205)(199,204)(200,210)(201,209)(202,208)(203,207)(211,220)(212,219)(213,218)(214,224)(215,223)(216,222)(217,221), (1,195,13,188)(2,196,14,189)(3,190,8,183)(4,191,9,184)(5,192,10,185)(6,193,11,186)(7,194,12,187)(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), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,106,71,92)(58,107,72,93)(59,108,73,94)(60,109,74,95)(61,110,75,96)(62,111,76,97)(63,112,77,98)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91)(113,155,127,141)(114,156,128,142)(115,157,129,143)(116,158,130,144)(117,159,131,145)(118,160,132,146)(119,161,133,147)(120,162,134,148)(121,163,135,149)(122,164,136,150)(123,165,137,151)(124,166,138,152)(125,167,139,153)(126,168,140,154)(169,218,183,204)(170,219,184,205)(171,220,185,206)(172,221,186,207)(173,222,187,208)(174,223,188,209)(175,224,189,210)(176,211,190,197)(177,212,191,198)(178,213,192,199)(179,214,193,200)(180,215,194,201)(181,216,195,202)(182,217,196,203) );
G=PermutationGroup([[(1,125),(2,126),(3,120),(4,121),(5,122),(6,123),(7,124),(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)], [(1,2,3,4,5,6,7),(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,14),(7,13),(15,24),(16,23),(17,22),(18,28),(19,27),(20,26),(21,25),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109),(113,122),(114,121),(115,120),(116,126),(117,125),(118,124),(119,123),(127,136),(128,135),(129,134),(130,140),(131,139),(132,138),(133,137),(141,150),(142,149),(143,148),(144,154),(145,153),(146,152),(147,151),(155,164),(156,163),(157,162),(158,168),(159,167),(160,166),(161,165),(169,178),(170,177),(171,176),(172,182),(173,181),(174,180),(175,179),(183,192),(184,191),(185,190),(186,196),(187,195),(188,194),(189,193),(197,206),(198,205),(199,204),(200,210),(201,209),(202,208),(203,207),(211,220),(212,219),(213,218),(214,224),(215,223),(216,222),(217,221)], [(1,195,13,188),(2,196,14,189),(3,190,8,183),(4,191,9,184),(5,192,10,185),(6,193,11,186),(7,194,12,187),(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)], [(1,48,20,34),(2,49,21,35),(3,43,15,29),(4,44,16,30),(5,45,17,31),(6,46,18,32),(7,47,19,33),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,106,71,92),(58,107,72,93),(59,108,73,94),(60,109,74,95),(61,110,75,96),(62,111,76,97),(63,112,77,98),(64,99,78,85),(65,100,79,86),(66,101,80,87),(67,102,81,88),(68,103,82,89),(69,104,83,90),(70,105,84,91),(113,155,127,141),(114,156,128,142),(115,157,129,143),(116,158,130,144),(117,159,131,145),(118,160,132,146),(119,161,133,147),(120,162,134,148),(121,163,135,149),(122,164,136,150),(123,165,137,151),(124,166,138,152),(125,167,139,153),(126,168,140,154),(169,218,183,204),(170,219,184,205),(171,220,185,206),(172,221,186,207),(173,222,187,208),(174,223,188,209),(175,224,189,210),(176,211,190,197),(177,212,191,198),(178,213,192,199),(179,214,193,200),(180,215,194,201),(181,216,195,202),(182,217,196,203)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4L | 4M | ··· | 4X | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D7 | D14 | D14 | C4×D7 | D4×D7 | Q8×D7 |
| kernel | C2×D7×C4⋊C4 | D7×C4⋊C4 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C14×C4⋊C4 | D7×C22×C4 | C2×C4×D7 | C22×D7 | C22×D7 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 2 | 1 | 1 | 3 | 16 | 4 | 4 | 3 | 12 | 9 | 24 | 6 | 6 |
Matrix representation of C2×D7×C4⋊C4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 21 | 1 | 0 | 0 | 0 | 0 |
| 27 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 26 | 3 | 0 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[21,27,0,0,0,0,1,11,0,0,0,0,0,0,7,28,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,7,0,0,0,0,3,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,1] >;
C2×D7×C4⋊C4 in GAP, Magma, Sage, TeX
C_2\times D_7\times C_4\rtimes C_4
% in TeX
G:=Group("C2xD7xC4:C4"); // GroupNames label
G:=SmallGroup(448,954);
// by ID
G=gap.SmallGroup(448,954);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,297,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^4=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations