metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.13D28, C22⋊C8⋊7D7, C56⋊1C4⋊5C2, C8⋊Dic7⋊9C2, (C2×C8).110D14, C28⋊7D4.8C2, (C2×C28).242D4, (C2×C4).120D28, C2.D56⋊11C2, C14.10(C4○D8), (C22×C4).87D14, (C22×C14).57D4, C28.284(C4○D4), C2.15(C8⋊D14), C14.12(C8⋊C22), (C2×C56).121C22, (C2×C28).747C23, (C2×D28).12C22, C22.110(C2×D28), C7⋊1(C23.19D4), C4⋊Dic7.14C22, C4.108(D4⋊2D7), C2.12(D56⋊7C2), C23.21D14⋊1C2, (C22×C28).98C22, C14.19(C22.D4), C2.15(C22.D28), (C7×C22⋊C8)⋊9C2, (C2×C14).130(C2×D4), (C2×C4).692(C22×D7), SmallGroup(448,271)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.13D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, ab=ba, eae-1=ac=ca, dad-1=abc, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd27 >
Subgroups: 636 in 106 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C23.19D4, C4×Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C2×C56, C2×D28, C2×C7⋊D4, C22×C28, C8⋊Dic7, C56⋊1C4, C2.D56, C7×C22⋊C8, C23.21D14, C28⋊7D4, C23.13D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8⋊C22, D28, C22×D7, C23.19D4, C2×D28, D4⋊2D7, C22.D28, D56⋊7C2, C8⋊D14, C23.13D28
►On 224 points
Generators in S224
(2 170)(4 172)(6 174)(8 176)(10 178)(12 180)(14 182)(16 184)(18 186)(20 188)(22 190)(24 192)(26 194)(28 196)(30 198)(32 200)(34 202)(36 204)(38 206)(40 208)(42 210)(44 212)(46 214)(48 216)(50 218)(52 220)(54 222)(56 224)(57 85)(58 140)(59 87)(60 142)(61 89)(62 144)(63 91)(64 146)(65 93)(66 148)(67 95)(68 150)(69 97)(70 152)(71 99)(72 154)(73 101)(74 156)(75 103)(76 158)(77 105)(78 160)(79 107)(80 162)(81 109)(82 164)(83 111)(84 166)(86 168)(88 114)(90 116)(92 118)(94 120)(96 122)(98 124)(100 126)(102 128)(104 130)(106 132)(108 134)(110 136)(112 138)(113 141)(115 143)(117 145)(119 147)(121 149)(123 151)(125 153)(127 155)(129 157)(131 159)(133 161)(135 163)(137 165)(139 167)
(1 197)(2 198)(3 199)(4 200)(5 201)(6 202)(7 203)(8 204)(9 205)(10 206)(11 207)(12 208)(13 209)(14 210)(15 211)(16 212)(17 213)(18 214)(19 215)(20 216)(21 217)(22 218)(23 219)(24 220)(25 221)(26 222)(27 223)(28 224)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 161)(80 162)(81 163)(82 164)(83 165)(84 166)(85 167)(86 168)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)
(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 99 169 153)(2 152 170 98)(3 97 171 151)(4 150 172 96)(5 95 173 149)(6 148 174 94)(7 93 175 147)(8 146 176 92)(9 91 177 145)(10 144 178 90)(11 89 179 143)(12 142 180 88)(13 87 181 141)(14 140 182 86)(15 85 183 139)(16 138 184 84)(17 83 185 137)(18 136 186 82)(19 81 187 135)(20 134 188 80)(21 79 189 133)(22 132 190 78)(23 77 191 131)(24 130 192 76)(25 75 193 129)(26 128 194 74)(27 73 195 127)(28 126 196 72)(29 71 197 125)(30 124 198 70)(31 69 199 123)(32 122 200 68)(33 67 201 121)(34 120 202 66)(35 65 203 119)(36 118 204 64)(37 63 205 117)(38 116 206 62)(39 61 207 115)(40 114 208 60)(41 59 209 113)(42 168 210 58)(43 57 211 167)(44 166 212 112)(45 111 213 165)(46 164 214 110)(47 109 215 163)(48 162 216 108)(49 107 217 161)(50 160 218 106)(51 105 219 159)(52 158 220 104)(53 103 221 157)(54 156 222 102)(55 101 223 155)(56 154 224 100)
G:=sub<Sym(224)| (2,170)(4,172)(6,174)(8,176)(10,178)(12,180)(14,182)(16,184)(18,186)(20,188)(22,190)(24,192)(26,194)(28,196)(30,198)(32,200)(34,202)(36,204)(38,206)(40,208)(42,210)(44,212)(46,214)(48,216)(50,218)(52,220)(54,222)(56,224)(57,85)(58,140)(59,87)(60,142)(61,89)(62,144)(63,91)(64,146)(65,93)(66,148)(67,95)(68,150)(69,97)(70,152)(71,99)(72,154)(73,101)(74,156)(75,103)(76,158)(77,105)(78,160)(79,107)(80,162)(81,109)(82,164)(83,111)(84,166)(86,168)(88,114)(90,116)(92,118)(94,120)(96,122)(98,124)(100,126)(102,128)(104,130)(106,132)(108,134)(110,136)(112,138)(113,141)(115,143)(117,145)(119,147)(121,149)(123,151)(125,153)(127,155)(129,157)(131,159)(133,161)(135,163)(137,165)(139,167), (1,197)(2,198)(3,199)(4,200)(5,201)(6,202)(7,203)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,218)(23,219)(24,220)(25,221)(26,222)(27,223)(28,224)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,161)(80,162)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138), (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), (1,99,169,153)(2,152,170,98)(3,97,171,151)(4,150,172,96)(5,95,173,149)(6,148,174,94)(7,93,175,147)(8,146,176,92)(9,91,177,145)(10,144,178,90)(11,89,179,143)(12,142,180,88)(13,87,181,141)(14,140,182,86)(15,85,183,139)(16,138,184,84)(17,83,185,137)(18,136,186,82)(19,81,187,135)(20,134,188,80)(21,79,189,133)(22,132,190,78)(23,77,191,131)(24,130,192,76)(25,75,193,129)(26,128,194,74)(27,73,195,127)(28,126,196,72)(29,71,197,125)(30,124,198,70)(31,69,199,123)(32,122,200,68)(33,67,201,121)(34,120,202,66)(35,65,203,119)(36,118,204,64)(37,63,205,117)(38,116,206,62)(39,61,207,115)(40,114,208,60)(41,59,209,113)(42,168,210,58)(43,57,211,167)(44,166,212,112)(45,111,213,165)(46,164,214,110)(47,109,215,163)(48,162,216,108)(49,107,217,161)(50,160,218,106)(51,105,219,159)(52,158,220,104)(53,103,221,157)(54,156,222,102)(55,101,223,155)(56,154,224,100)>;
G:=Group( (2,170)(4,172)(6,174)(8,176)(10,178)(12,180)(14,182)(16,184)(18,186)(20,188)(22,190)(24,192)(26,194)(28,196)(30,198)(32,200)(34,202)(36,204)(38,206)(40,208)(42,210)(44,212)(46,214)(48,216)(50,218)(52,220)(54,222)(56,224)(57,85)(58,140)(59,87)(60,142)(61,89)(62,144)(63,91)(64,146)(65,93)(66,148)(67,95)(68,150)(69,97)(70,152)(71,99)(72,154)(73,101)(74,156)(75,103)(76,158)(77,105)(78,160)(79,107)(80,162)(81,109)(82,164)(83,111)(84,166)(86,168)(88,114)(90,116)(92,118)(94,120)(96,122)(98,124)(100,126)(102,128)(104,130)(106,132)(108,134)(110,136)(112,138)(113,141)(115,143)(117,145)(119,147)(121,149)(123,151)(125,153)(127,155)(129,157)(131,159)(133,161)(135,163)(137,165)(139,167), (1,197)(2,198)(3,199)(4,200)(5,201)(6,202)(7,203)(8,204)(9,205)(10,206)(11,207)(12,208)(13,209)(14,210)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,218)(23,219)(24,220)(25,221)(26,222)(27,223)(28,224)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,161)(80,162)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138), (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), (1,99,169,153)(2,152,170,98)(3,97,171,151)(4,150,172,96)(5,95,173,149)(6,148,174,94)(7,93,175,147)(8,146,176,92)(9,91,177,145)(10,144,178,90)(11,89,179,143)(12,142,180,88)(13,87,181,141)(14,140,182,86)(15,85,183,139)(16,138,184,84)(17,83,185,137)(18,136,186,82)(19,81,187,135)(20,134,188,80)(21,79,189,133)(22,132,190,78)(23,77,191,131)(24,130,192,76)(25,75,193,129)(26,128,194,74)(27,73,195,127)(28,126,196,72)(29,71,197,125)(30,124,198,70)(31,69,199,123)(32,122,200,68)(33,67,201,121)(34,120,202,66)(35,65,203,119)(36,118,204,64)(37,63,205,117)(38,116,206,62)(39,61,207,115)(40,114,208,60)(41,59,209,113)(42,168,210,58)(43,57,211,167)(44,166,212,112)(45,111,213,165)(46,164,214,110)(47,109,215,163)(48,162,216,108)(49,107,217,161)(50,160,218,106)(51,105,219,159)(52,158,220,104)(53,103,221,157)(54,156,222,102)(55,101,223,155)(56,154,224,100) );
G=PermutationGroup([[(2,170),(4,172),(6,174),(8,176),(10,178),(12,180),(14,182),(16,184),(18,186),(20,188),(22,190),(24,192),(26,194),(28,196),(30,198),(32,200),(34,202),(36,204),(38,206),(40,208),(42,210),(44,212),(46,214),(48,216),(50,218),(52,220),(54,222),(56,224),(57,85),(58,140),(59,87),(60,142),(61,89),(62,144),(63,91),(64,146),(65,93),(66,148),(67,95),(68,150),(69,97),(70,152),(71,99),(72,154),(73,101),(74,156),(75,103),(76,158),(77,105),(78,160),(79,107),(80,162),(81,109),(82,164),(83,111),(84,166),(86,168),(88,114),(90,116),(92,118),(94,120),(96,122),(98,124),(100,126),(102,128),(104,130),(106,132),(108,134),(110,136),(112,138),(113,141),(115,143),(117,145),(119,147),(121,149),(123,151),(125,153),(127,155),(129,157),(131,159),(133,161),(135,163),(137,165),(139,167)], [(1,197),(2,198),(3,199),(4,200),(5,201),(6,202),(7,203),(8,204),(9,205),(10,206),(11,207),(12,208),(13,209),(14,210),(15,211),(16,212),(17,213),(18,214),(19,215),(20,216),(21,217),(22,218),(23,219),(24,220),(25,221),(26,222),(27,223),(28,224),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,193),(54,194),(55,195),(56,196),(57,139),(58,140),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,154),(73,155),(74,156),(75,157),(76,158),(77,159),(78,160),(79,161),(80,162),(81,163),(82,164),(83,165),(84,166),(85,167),(86,168),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138)], [(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)], [(1,99,169,153),(2,152,170,98),(3,97,171,151),(4,150,172,96),(5,95,173,149),(6,148,174,94),(7,93,175,147),(8,146,176,92),(9,91,177,145),(10,144,178,90),(11,89,179,143),(12,142,180,88),(13,87,181,141),(14,140,182,86),(15,85,183,139),(16,138,184,84),(17,83,185,137),(18,136,186,82),(19,81,187,135),(20,134,188,80),(21,79,189,133),(22,132,190,78),(23,77,191,131),(24,130,192,76),(25,75,193,129),(26,128,194,74),(27,73,195,127),(28,126,196,72),(29,71,197,125),(30,124,198,70),(31,69,199,123),(32,122,200,68),(33,67,201,121),(34,120,202,66),(35,65,203,119),(36,118,204,64),(37,63,205,117),(38,116,206,62),(39,61,207,115),(40,114,208,60),(41,59,209,113),(42,168,210,58),(43,57,211,167),(44,166,212,112),(45,111,213,165),(46,164,214,110),(47,109,215,163),(48,162,216,108),(49,107,217,161),(50,160,218,106),(51,105,219,159),(52,158,220,104),(53,103,221,157),(54,156,222,102),(55,101,223,155),(56,154,224,100)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 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 | 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 | 4 | 56 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C4○D8 | D28 | D28 | D56⋊7C2 | C8⋊C22 | D4⋊2D7 | C8⋊D14 |
| kernel | C23.13D28 | C8⋊Dic7 | C56⋊1C4 | C2.D56 | C7×C22⋊C8 | C23.21D14 | C28⋊7D4 | C2×C28 | C22×C14 | C22⋊C8 | C28 | C2×C8 | C22×C4 | C14 | C2×C4 | C23 | C2 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 3 | 4 | 6 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of C23.13D28 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 46 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 29 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 41 | 0 | 0 | 0 |
| 66 | 102 | 0 | 0 |
| 0 | 0 | 18 | 65 |
| 0 | 0 | 108 | 95 |
| 63 | 12 | 0 | 0 |
| 46 | 50 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(113))| [1,46,0,0,0,112,0,0,0,0,1,29,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[41,66,0,0,0,102,0,0,0,0,18,108,0,0,65,95],[63,46,0,0,12,50,0,0,0,0,15,0,0,0,0,15] >;
C23.13D28 in GAP, Magma, Sage, TeX
C_2^3._{13}D_{28} % in TeX
G:=Group("C2^3.13D28"); // GroupNames label
G:=SmallGroup(448,271);
// by ID
G=gap.SmallGroup(448,271);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,555,142,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,d*a*d^-1=a*b*c,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