metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.46D28, (C2×C28).166D4, (C2×C8).187D14, C28.415(C2×D4), (C2×C4).148D28, C4.11(D14⋊C4), (C2×Dic14)⋊14C4, C22.55(C2×D28), (C22×C14).99D4, C28.44D4⋊39C2, C28.24(C22⋊C4), C28.114(C22×C4), (C2×C56).317C22, (C2×C28).771C23, Dic14.25(C2×C4), C2.3(C8.D14), (C22×C4).132D14, C7⋊3(C23.38D4), (C2×M4(2)).14D7, C22.26(D14⋊C4), C14.19(C8.C22), (C14×M4(2)).25C2, C4⋊Dic7.283C22, (C22×C28).179C22, (C22×Dic14).14C2, (C2×Dic14).220C22, C23.21D14.16C2, C4.72(C2×C4×D7), (C2×C4).48(C4×D7), (C2×C28).99(C2×C4), C2.26(C2×D14⋊C4), C4.108(C2×C7⋊D4), (C2×C14).161(C2×D4), (C2×C4).75(C7⋊D4), C14.54(C2×C22⋊C4), (C2×C4).719(C22×D7), (C2×C14).18(C22⋊C4), SmallGroup(448,654)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.46D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=e2=c, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd27 >
Subgroups: 676 in 150 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C56, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.38D4, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C28.44D4, C23.21D14, C14×M4(2), C22×Dic14, C23.46D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C8.C22, C4×D7, D28, C7⋊D4, C22×D7, C23.38D4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C8.D14, C2×D14⋊C4, C23.46D28
►On 224 points
Generators in S224
(1 213)(2 186)(3 215)(4 188)(5 217)(6 190)(7 219)(8 192)(9 221)(10 194)(11 223)(12 196)(13 169)(14 198)(15 171)(16 200)(17 173)(18 202)(19 175)(20 204)(21 177)(22 206)(23 179)(24 208)(25 181)(26 210)(27 183)(28 212)(29 185)(30 214)(31 187)(32 216)(33 189)(34 218)(35 191)(36 220)(37 193)(38 222)(39 195)(40 224)(41 197)(42 170)(43 199)(44 172)(45 201)(46 174)(47 203)(48 176)(49 205)(50 178)(51 207)(52 180)(53 209)(54 182)(55 211)(56 184)(57 130)(58 159)(59 132)(60 161)(61 134)(62 163)(63 136)(64 165)(65 138)(66 167)(67 140)(68 113)(69 142)(70 115)(71 144)(72 117)(73 146)(74 119)(75 148)(76 121)(77 150)(78 123)(79 152)(80 125)(81 154)(82 127)(83 156)(84 129)(85 158)(86 131)(87 160)(88 133)(89 162)(90 135)(91 164)(92 137)(93 166)(94 139)(95 168)(96 141)(97 114)(98 143)(99 116)(100 145)(101 118)(102 147)(103 120)(104 149)(105 122)(106 151)(107 124)(108 153)(109 126)(110 155)(111 128)(112 157)
(1 213)(2 214)(3 215)(4 216)(5 217)(6 218)(7 219)(8 220)(9 221)(10 222)(11 223)(12 224)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 205)(50 206)(51 207)(52 208)(53 209)(54 210)(55 211)(56 212)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 142)(70 143)(71 144)(72 145)(73 146)(74 147)(75 148)(76 149)(77 150)(78 151)(79 152)(80 153)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 161)(89 162)(90 163)(91 164)(92 165)(93 166)(94 167)(95 168)(96 113)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 129)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(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)
(1 126 29 154)(2 80 30 108)(3 124 31 152)(4 78 32 106)(5 122 33 150)(6 76 34 104)(7 120 35 148)(8 74 36 102)(9 118 37 146)(10 72 38 100)(11 116 39 144)(12 70 40 98)(13 114 41 142)(14 68 42 96)(15 168 43 140)(16 66 44 94)(17 166 45 138)(18 64 46 92)(19 164 47 136)(20 62 48 90)(21 162 49 134)(22 60 50 88)(23 160 51 132)(24 58 52 86)(25 158 53 130)(26 112 54 84)(27 156 55 128)(28 110 56 82)(57 181 85 209)(59 179 87 207)(61 177 89 205)(63 175 91 203)(65 173 93 201)(67 171 95 199)(69 169 97 197)(71 223 99 195)(73 221 101 193)(75 219 103 191)(77 217 105 189)(79 215 107 187)(81 213 109 185)(83 211 111 183)(113 170 141 198)(115 224 143 196)(117 222 145 194)(119 220 147 192)(121 218 149 190)(123 216 151 188)(125 214 153 186)(127 212 155 184)(129 210 157 182)(131 208 159 180)(133 206 161 178)(135 204 163 176)(137 202 165 174)(139 200 167 172)
G:=sub<Sym(224)| (1,213)(2,186)(3,215)(4,188)(5,217)(6,190)(7,219)(8,192)(9,221)(10,194)(11,223)(12,196)(13,169)(14,198)(15,171)(16,200)(17,173)(18,202)(19,175)(20,204)(21,177)(22,206)(23,179)(24,208)(25,181)(26,210)(27,183)(28,212)(29,185)(30,214)(31,187)(32,216)(33,189)(34,218)(35,191)(36,220)(37,193)(38,222)(39,195)(40,224)(41,197)(42,170)(43,199)(44,172)(45,201)(46,174)(47,203)(48,176)(49,205)(50,178)(51,207)(52,180)(53,209)(54,182)(55,211)(56,184)(57,130)(58,159)(59,132)(60,161)(61,134)(62,163)(63,136)(64,165)(65,138)(66,167)(67,140)(68,113)(69,142)(70,115)(71,144)(72,117)(73,146)(74,119)(75,148)(76,121)(77,150)(78,123)(79,152)(80,125)(81,154)(82,127)(83,156)(84,129)(85,158)(86,131)(87,160)(88,133)(89,162)(90,135)(91,164)(92,137)(93,166)(94,139)(95,168)(96,141)(97,114)(98,143)(99,116)(100,145)(101,118)(102,147)(103,120)(104,149)(105,122)(106,151)(107,124)(108,153)(109,126)(110,155)(111,128)(112,157), (1,213)(2,214)(3,215)(4,216)(5,217)(6,218)(7,219)(8,220)(9,221)(10,222)(11,223)(12,224)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,181)(26,182)(27,183)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,211)(56,212)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,126,29,154)(2,80,30,108)(3,124,31,152)(4,78,32,106)(5,122,33,150)(6,76,34,104)(7,120,35,148)(8,74,36,102)(9,118,37,146)(10,72,38,100)(11,116,39,144)(12,70,40,98)(13,114,41,142)(14,68,42,96)(15,168,43,140)(16,66,44,94)(17,166,45,138)(18,64,46,92)(19,164,47,136)(20,62,48,90)(21,162,49,134)(22,60,50,88)(23,160,51,132)(24,58,52,86)(25,158,53,130)(26,112,54,84)(27,156,55,128)(28,110,56,82)(57,181,85,209)(59,179,87,207)(61,177,89,205)(63,175,91,203)(65,173,93,201)(67,171,95,199)(69,169,97,197)(71,223,99,195)(73,221,101,193)(75,219,103,191)(77,217,105,189)(79,215,107,187)(81,213,109,185)(83,211,111,183)(113,170,141,198)(115,224,143,196)(117,222,145,194)(119,220,147,192)(121,218,149,190)(123,216,151,188)(125,214,153,186)(127,212,155,184)(129,210,157,182)(131,208,159,180)(133,206,161,178)(135,204,163,176)(137,202,165,174)(139,200,167,172)>;
G:=Group( (1,213)(2,186)(3,215)(4,188)(5,217)(6,190)(7,219)(8,192)(9,221)(10,194)(11,223)(12,196)(13,169)(14,198)(15,171)(16,200)(17,173)(18,202)(19,175)(20,204)(21,177)(22,206)(23,179)(24,208)(25,181)(26,210)(27,183)(28,212)(29,185)(30,214)(31,187)(32,216)(33,189)(34,218)(35,191)(36,220)(37,193)(38,222)(39,195)(40,224)(41,197)(42,170)(43,199)(44,172)(45,201)(46,174)(47,203)(48,176)(49,205)(50,178)(51,207)(52,180)(53,209)(54,182)(55,211)(56,184)(57,130)(58,159)(59,132)(60,161)(61,134)(62,163)(63,136)(64,165)(65,138)(66,167)(67,140)(68,113)(69,142)(70,115)(71,144)(72,117)(73,146)(74,119)(75,148)(76,121)(77,150)(78,123)(79,152)(80,125)(81,154)(82,127)(83,156)(84,129)(85,158)(86,131)(87,160)(88,133)(89,162)(90,135)(91,164)(92,137)(93,166)(94,139)(95,168)(96,141)(97,114)(98,143)(99,116)(100,145)(101,118)(102,147)(103,120)(104,149)(105,122)(106,151)(107,124)(108,153)(109,126)(110,155)(111,128)(112,157), (1,213)(2,214)(3,215)(4,216)(5,217)(6,218)(7,219)(8,220)(9,221)(10,222)(11,223)(12,224)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,181)(26,182)(27,183)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,211)(56,212)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,142)(70,143)(71,144)(72,145)(73,146)(74,147)(75,148)(76,149)(77,150)(78,151)(79,152)(80,153)(81,154)(82,155)(83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,165)(93,166)(94,167)(95,168)(96,113)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,126,29,154)(2,80,30,108)(3,124,31,152)(4,78,32,106)(5,122,33,150)(6,76,34,104)(7,120,35,148)(8,74,36,102)(9,118,37,146)(10,72,38,100)(11,116,39,144)(12,70,40,98)(13,114,41,142)(14,68,42,96)(15,168,43,140)(16,66,44,94)(17,166,45,138)(18,64,46,92)(19,164,47,136)(20,62,48,90)(21,162,49,134)(22,60,50,88)(23,160,51,132)(24,58,52,86)(25,158,53,130)(26,112,54,84)(27,156,55,128)(28,110,56,82)(57,181,85,209)(59,179,87,207)(61,177,89,205)(63,175,91,203)(65,173,93,201)(67,171,95,199)(69,169,97,197)(71,223,99,195)(73,221,101,193)(75,219,103,191)(77,217,105,189)(79,215,107,187)(81,213,109,185)(83,211,111,183)(113,170,141,198)(115,224,143,196)(117,222,145,194)(119,220,147,192)(121,218,149,190)(123,216,151,188)(125,214,153,186)(127,212,155,184)(129,210,157,182)(131,208,159,180)(133,206,161,178)(135,204,163,176)(137,202,165,174)(139,200,167,172) );
G=PermutationGroup([[(1,213),(2,186),(3,215),(4,188),(5,217),(6,190),(7,219),(8,192),(9,221),(10,194),(11,223),(12,196),(13,169),(14,198),(15,171),(16,200),(17,173),(18,202),(19,175),(20,204),(21,177),(22,206),(23,179),(24,208),(25,181),(26,210),(27,183),(28,212),(29,185),(30,214),(31,187),(32,216),(33,189),(34,218),(35,191),(36,220),(37,193),(38,222),(39,195),(40,224),(41,197),(42,170),(43,199),(44,172),(45,201),(46,174),(47,203),(48,176),(49,205),(50,178),(51,207),(52,180),(53,209),(54,182),(55,211),(56,184),(57,130),(58,159),(59,132),(60,161),(61,134),(62,163),(63,136),(64,165),(65,138),(66,167),(67,140),(68,113),(69,142),(70,115),(71,144),(72,117),(73,146),(74,119),(75,148),(76,121),(77,150),(78,123),(79,152),(80,125),(81,154),(82,127),(83,156),(84,129),(85,158),(86,131),(87,160),(88,133),(89,162),(90,135),(91,164),(92,137),(93,166),(94,139),(95,168),(96,141),(97,114),(98,143),(99,116),(100,145),(101,118),(102,147),(103,120),(104,149),(105,122),(106,151),(107,124),(108,153),(109,126),(110,155),(111,128),(112,157)], [(1,213),(2,214),(3,215),(4,216),(5,217),(6,218),(7,219),(8,220),(9,221),(10,222),(11,223),(12,224),(13,169),(14,170),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,177),(22,178),(23,179),(24,180),(25,181),(26,182),(27,183),(28,184),(29,185),(30,186),(31,187),(32,188),(33,189),(34,190),(35,191),(36,192),(37,193),(38,194),(39,195),(40,196),(41,197),(42,198),(43,199),(44,200),(45,201),(46,202),(47,203),(48,204),(49,205),(50,206),(51,207),(52,208),(53,209),(54,210),(55,211),(56,212),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,141),(69,142),(70,143),(71,144),(72,145),(73,146),(74,147),(75,148),(76,149),(77,150),(78,151),(79,152),(80,153),(81,154),(82,155),(83,156),(84,157),(85,158),(86,159),(87,160),(88,161),(89,162),(90,163),(91,164),(92,165),(93,166),(94,167),(95,168),(96,113),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,121),(105,122),(106,123),(107,124),(108,125),(109,126),(110,127),(111,128),(112,129)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(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)], [(1,126,29,154),(2,80,30,108),(3,124,31,152),(4,78,32,106),(5,122,33,150),(6,76,34,104),(7,120,35,148),(8,74,36,102),(9,118,37,146),(10,72,38,100),(11,116,39,144),(12,70,40,98),(13,114,41,142),(14,68,42,96),(15,168,43,140),(16,66,44,94),(17,166,45,138),(18,64,46,92),(19,164,47,136),(20,62,48,90),(21,162,49,134),(22,60,50,88),(23,160,51,132),(24,58,52,86),(25,158,53,130),(26,112,54,84),(27,156,55,128),(28,110,56,82),(57,181,85,209),(59,179,87,207),(61,177,89,205),(63,175,91,203),(65,173,93,201),(67,171,95,199),(69,169,97,197),(71,223,99,195),(73,221,101,193),(75,219,103,191),(77,217,105,189),(79,215,107,187),(81,213,109,185),(83,211,111,183),(113,170,141,198),(115,224,143,196),(117,222,145,194),(119,220,147,192),(121,218,149,190),(123,216,151,188),(125,214,153,186),(127,212,155,184),(129,210,157,182),(131,208,159,180),(133,206,161,178),(135,204,163,176),(137,202,165,174),(139,200,167,172)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D28 | C8.C22 | C8.D14 |
| kernel | C23.46D28 | C28.44D4 | C23.21D14 | C14×M4(2) | C22×Dic14 | C2×Dic14 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 6 | 3 | 12 | 6 | 12 | 6 | 2 | 12 |
Matrix representation of C23.46D28 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 39 | 105 | 0 | 0 | 0 | 0 |
| 77 | 74 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 80 | 104 |
| 0 | 0 | 0 | 0 | 9 | 112 |
| 0 | 0 | 4 | 81 | 0 | 0 |
| 0 | 0 | 32 | 55 | 0 | 0 |
| 85 | 6 | 0 | 0 | 0 | 0 |
| 39 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 63 | 71 | 0 | 0 |
| 0 | 0 | 73 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 42 |
| 0 | 0 | 0 | 0 | 32 | 89 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[39,77,0,0,0,0,105,74,0,0,0,0,0,0,0,0,4,32,0,0,0,0,81,55,0,0,80,9,0,0,0,0,104,112,0,0],[85,39,0,0,0,0,6,28,0,0,0,0,0,0,63,73,0,0,0,0,71,50,0,0,0,0,0,0,24,32,0,0,0,0,42,89] >;
C23.46D28 in GAP, Magma, Sage, TeX
C_2^3._{46}D_{28} % in TeX
G:=Group("C2^3.46D28"); // GroupNames label
G:=SmallGroup(448,654);
// by ID
G=gap.SmallGroup(448,654);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,254,387,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=e^2=c,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;
// generators/relations