direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23×Dic6, C6.1C25, C24.91D6, C12.64C24, Dic3.1C24, C3⋊1(Q8×C23), (C22×C6)⋊8Q8, C6⋊1(C22×Q8), C2.3(S3×C24), C4.61(S3×C23), (C23×C4).19S3, (C23×C12).14C2, (C2×C6).323C24, (C22×C4).466D6, (C2×C12).790C23, C22.51(S3×C23), (C22×C6).430C23, C23.354(C22×S3), (C23×C6).113C22, (C23×Dic3).11C2, (C22×C12).531C22, (C2×Dic3).295C23, (C22×Dic3).238C22, (C2×C6)⋊7(C2×Q8), (C2×C4).741(C22×S3), SmallGroup(192,1510)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23×Dic6
G = < a,b,c,d,e | a2=b2=c2=d12=1, e2=d6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 1464 in 850 conjugacy classes, 543 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C2×C6, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C22×C6, C23×C4, C23×C4, C22×Q8, C2×Dic6, C22×Dic3, C22×C12, C23×C6, Q8×C23, C22×Dic6, C23×Dic3, C23×C12, C23×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, Dic6, C22×S3, C22×Q8, C25, C2×Dic6, S3×C23, Q8×C23, C22×Dic6, S3×C24, C23×Dic6
►Regular action on 192 points
Generators in S192
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 49)(10 50)(11 51)(12 52)(13 93)(14 94)(15 95)(16 96)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 69)(26 70)(27 71)(28 72)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 121)(48 122)(73 104)(74 105)(75 106)(76 107)(77 108)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(151 169)(152 170)(153 171)(154 172)(155 173)(156 174)(157 187)(158 188)(159 189)(160 190)(161 191)(162 192)(163 181)(164 182)(165 183)(166 184)(167 185)(168 186)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 44)(26 45)(27 46)(28 47)(29 48)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)(49 92)(50 93)(51 94)(52 95)(53 96)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 121)(73 169)(74 170)(75 171)(76 172)(77 173)(78 174)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(97 156)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 192)(110 181)(111 182)(112 183)(113 184)(114 185)(115 186)(116 187)(117 188)(118 189)(119 190)(120 191)(133 168)(134 157)(135 158)(136 159)(137 160)(138 161)(139 162)(140 163)(141 164)(142 165)(143 166)(144 167)
(1 132)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 128)(10 129)(11 130)(12 131)(13 68)(14 69)(15 70)(16 71)(17 72)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 94)(26 95)(27 96)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 56)(38 57)(39 58)(40 59)(41 60)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(97 168)(98 157)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 180)(110 169)(111 170)(112 171)(113 172)(114 173)(115 174)(116 175)(117 176)(118 177)(119 178)(120 179)(133 156)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 151)(141 152)(142 153)(143 154)(144 155)
(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)
(1 160 7 166)(2 159 8 165)(3 158 9 164)(4 157 10 163)(5 168 11 162)(6 167 12 161)(13 140 19 134)(14 139 20 133)(15 138 21 144)(16 137 22 143)(17 136 23 142)(18 135 24 141)(25 180 31 174)(26 179 32 173)(27 178 33 172)(28 177 34 171)(29 176 35 170)(30 175 36 169)(37 79 43 73)(38 78 44 84)(39 77 45 83)(40 76 46 82)(41 75 47 81)(42 74 48 80)(49 182 55 188)(50 181 56 187)(51 192 57 186)(52 191 58 185)(53 190 59 184)(54 189 60 183)(61 146 67 152)(62 145 68 151)(63 156 69 150)(64 155 70 149)(65 154 71 148)(66 153 72 147)(85 118 91 112)(86 117 92 111)(87 116 93 110)(88 115 94 109)(89 114 95 120)(90 113 96 119)(97 130 103 124)(98 129 104 123)(99 128 105 122)(100 127 106 121)(101 126 107 132)(102 125 108 131)
G:=sub<Sym(192)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,49)(10,50)(11,51)(12,52)(13,93)(14,94)(15,95)(16,96)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,69)(26,70)(27,71)(28,72)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,121)(48,122)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(151,169)(152,170)(153,171)(154,172)(155,173)(156,174)(157,187)(158,188)(159,189)(160,190)(161,191)(162,192)(163,181)(164,182)(165,183)(166,184)(167,185)(168,186), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,44)(26,45)(27,46)(28,47)(29,48)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43)(49,92)(50,93)(51,94)(52,95)(53,96)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,121)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(97,156)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,192)(110,181)(111,182)(112,183)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(133,168)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167), (1,132)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,68)(14,69)(15,70)(16,71)(17,72)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,94)(26,95)(27,96)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,56)(38,57)(39,58)(40,59)(41,60)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(97,168)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,180)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179)(133,156)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)(142,153)(143,154)(144,155), (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), (1,160,7,166)(2,159,8,165)(3,158,9,164)(4,157,10,163)(5,168,11,162)(6,167,12,161)(13,140,19,134)(14,139,20,133)(15,138,21,144)(16,137,22,143)(17,136,23,142)(18,135,24,141)(25,180,31,174)(26,179,32,173)(27,178,33,172)(28,177,34,171)(29,176,35,170)(30,175,36,169)(37,79,43,73)(38,78,44,84)(39,77,45,83)(40,76,46,82)(41,75,47,81)(42,74,48,80)(49,182,55,188)(50,181,56,187)(51,192,57,186)(52,191,58,185)(53,190,59,184)(54,189,60,183)(61,146,67,152)(62,145,68,151)(63,156,69,150)(64,155,70,149)(65,154,71,148)(66,153,72,147)(85,118,91,112)(86,117,92,111)(87,116,93,110)(88,115,94,109)(89,114,95,120)(90,113,96,119)(97,130,103,124)(98,129,104,123)(99,128,105,122)(100,127,106,121)(101,126,107,132)(102,125,108,131)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,49)(10,50)(11,51)(12,52)(13,93)(14,94)(15,95)(16,96)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,69)(26,70)(27,71)(28,72)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,121)(48,122)(73,104)(74,105)(75,106)(76,107)(77,108)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(151,169)(152,170)(153,171)(154,172)(155,173)(156,174)(157,187)(158,188)(159,189)(160,190)(161,191)(162,192)(163,181)(164,182)(165,183)(166,184)(167,185)(168,186), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,44)(26,45)(27,46)(28,47)(29,48)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43)(49,92)(50,93)(51,94)(52,95)(53,96)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,121)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(97,156)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,192)(110,181)(111,182)(112,183)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(133,168)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167), (1,132)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,68)(14,69)(15,70)(16,71)(17,72)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,94)(26,95)(27,96)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,56)(38,57)(39,58)(40,59)(41,60)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(73,181)(74,182)(75,183)(76,184)(77,185)(78,186)(79,187)(80,188)(81,189)(82,190)(83,191)(84,192)(97,168)(98,157)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,180)(110,169)(111,170)(112,171)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179)(133,156)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)(142,153)(143,154)(144,155), (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), (1,160,7,166)(2,159,8,165)(3,158,9,164)(4,157,10,163)(5,168,11,162)(6,167,12,161)(13,140,19,134)(14,139,20,133)(15,138,21,144)(16,137,22,143)(17,136,23,142)(18,135,24,141)(25,180,31,174)(26,179,32,173)(27,178,33,172)(28,177,34,171)(29,176,35,170)(30,175,36,169)(37,79,43,73)(38,78,44,84)(39,77,45,83)(40,76,46,82)(41,75,47,81)(42,74,48,80)(49,182,55,188)(50,181,56,187)(51,192,57,186)(52,191,58,185)(53,190,59,184)(54,189,60,183)(61,146,67,152)(62,145,68,151)(63,156,69,150)(64,155,70,149)(65,154,71,148)(66,153,72,147)(85,118,91,112)(86,117,92,111)(87,116,93,110)(88,115,94,109)(89,114,95,120)(90,113,96,119)(97,130,103,124)(98,129,104,123)(99,128,105,122)(100,127,106,121)(101,126,107,132)(102,125,108,131) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,49),(10,50),(11,51),(12,52),(13,93),(14,94),(15,95),(16,96),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,69),(26,70),(27,71),(28,72),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,123),(38,124),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,121),(48,122),(73,104),(74,105),(75,106),(76,107),(77,108),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(109,139),(110,140),(111,141),(112,142),(113,143),(114,144),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(151,169),(152,170),(153,171),(154,172),(155,173),(156,174),(157,187),(158,188),(159,189),(160,190),(161,191),(162,192),(163,181),(164,182),(165,183),(166,184),(167,185),(168,186)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,44),(26,45),(27,46),(28,47),(29,48),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(36,43),(49,92),(50,93),(51,94),(52,95),(53,96),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,91),(61,122),(62,123),(63,124),(64,125),(65,126),(66,127),(67,128),(68,129),(69,130),(70,131),(71,132),(72,121),(73,169),(74,170),(75,171),(76,172),(77,173),(78,174),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(97,156),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,192),(110,181),(111,182),(112,183),(113,184),(114,185),(115,186),(116,187),(117,188),(118,189),(119,190),(120,191),(133,168),(134,157),(135,158),(136,159),(137,160),(138,161),(139,162),(140,163),(141,164),(142,165),(143,166),(144,167)], [(1,132),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,128),(10,129),(11,130),(12,131),(13,68),(14,69),(15,70),(16,71),(17,72),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,94),(26,95),(27,96),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,56),(38,57),(39,58),(40,59),(41,60),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(73,181),(74,182),(75,183),(76,184),(77,185),(78,186),(79,187),(80,188),(81,189),(82,190),(83,191),(84,192),(97,168),(98,157),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,180),(110,169),(111,170),(112,171),(113,172),(114,173),(115,174),(116,175),(117,176),(118,177),(119,178),(120,179),(133,156),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,151),(141,152),(142,153),(143,154),(144,155)], [(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)], [(1,160,7,166),(2,159,8,165),(3,158,9,164),(4,157,10,163),(5,168,11,162),(6,167,12,161),(13,140,19,134),(14,139,20,133),(15,138,21,144),(16,137,22,143),(17,136,23,142),(18,135,24,141),(25,180,31,174),(26,179,32,173),(27,178,33,172),(28,177,34,171),(29,176,35,170),(30,175,36,169),(37,79,43,73),(38,78,44,84),(39,77,45,83),(40,76,46,82),(41,75,47,81),(42,74,48,80),(49,182,55,188),(50,181,56,187),(51,192,57,186),(52,191,58,185),(53,190,59,184),(54,189,60,183),(61,146,67,152),(62,145,68,151),(63,156,69,150),(64,155,70,149),(65,154,71,148),(66,153,72,147),(85,118,91,112),(86,117,92,111),(87,116,93,110),(88,115,94,109),(89,114,95,120),(90,113,96,119),(97,130,103,124),(98,129,104,123),(99,128,105,122),(100,127,106,121),(101,126,107,132),(102,125,108,131)]])
72 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3 | 4A | ··· | 4H | 4I | ··· | 4X | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | Dic6 |
| kernel | C23×Dic6 | C22×Dic6 | C23×Dic3 | C23×C12 | C23×C4 | C22×C6 | C22×C4 | C24 | C23 |
| # reps | 1 | 28 | 2 | 1 | 1 | 8 | 14 | 1 | 16 |
Matrix representation of C23×Dic6 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 | 7 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 3 |
| 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,7],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,5,0,0,0,0,3,8] >;
C23×Dic6 in GAP, Magma, Sage, TeX
C_2^3\times {\rm Dic}_6 % in TeX
G:=Group("C2^3xDic6"); // GroupNames label
G:=SmallGroup(192,1510);
// by ID
G=gap.SmallGroup(192,1510);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=1,e^2=d^6,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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations