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G = C23×Dic6  order 192 = 26·3

Direct product of C23 and Dic6

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), C23.354(C22×S3), (C22×C6).430C23, (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

Derived series
C1 — C6 — C23×Dic6
Chief series
C1 — C3 — C6 — Dic3 — C2×Dic3 — C22×Dic3 — C23×Dic3 — C23×Dic6
Lower central
C3 — C6 — C23×Dic6

Subgroups: 1464 in 850 conjugacy classes, 543 normal (9 characteristic)
C1, C2, C2 [×14], C3, C4 [×8], C4 [×16], C22 [×35], C6, C6 [×14], C2×C4 [×28], C2×C4 [×56], Q8 [×64], C23 [×15], Dic3 [×16], C12 [×8], C2×C6 [×35], C22×C4 [×14], C22×C4 [×28], C2×Q8 [×112], C24, Dic6 [×64], C2×Dic3 [×56], C2×C12 [×28], C22×C6 [×15], C23×C4, C23×C4 [×2], C22×Q8 [×28], C2×Dic6 [×112], C22×Dic3 [×28], C22×C12 [×14], C23×C6, Q8×C23, C22×Dic6 [×28], C23×Dic3 [×2], C23×C12, C23×Dic6

Quotients:
C1, C2 [×31], C22 [×155], S3, Q8 [×8], C23 [×155], D6 [×15], C2×Q8 [×28], C24 [×31], Dic6 [×8], C22×S3 [×35], C22×Q8 [×14], C25, C2×Dic6 [×28], S3×C23 [×15], Q8×C23, C22×Dic6 [×14], S3×C24, C23×Dic6

Generators and relations
 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 >

Smallest permutation representation
►Regular action on 192 points
Generators in S192
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 73)(23 74)(24 75)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 85)(72 86)(97 173)(98 174)(99 175)(100 176)(101 177)(102 178)(103 179)(104 180)(105 169)(106 170)(107 171)(108 172)(109 155)(110 156)(111 145)(112 146)(113 147)(114 148)(115 149)(116 150)(117 151)(118 152)(119 153)(120 154)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)(157 181)(158 182)(159 183)(160 184)(161 185)(162 186)(163 187)(164 188)(165 189)(166 190)(167 191)(168 192)

(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 153)(14 154)(15 155)(16 156)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 58)(26 59)(27 60)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)(73 116)(74 117)(75 118)(76 119)(77 120)(78 109)(79 110)(80 111)(81 112)(82 113)(83 114)(84 115)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 133)(94 134)(95 135)(96 136)(97 190)(98 191)(99 192)(100 181)(101 182)(102 183)(103 184)(104 185)(105 186)(106 187)(107 188)(108 189)(157 176)(158 177)(159 178)(160 179)(161 180)(162 169)(163 170)(164 171)(165 172)(166 173)(167 174)(168 175)

(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 85)(12 86)(13 159)(14 160)(15 161)(16 162)(17 163)(18 164)(19 165)(20 166)(21 167)(22 168)(23 157)(24 158)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 61)(35 62)(36 63)(37 141)(38 142)(39 143)(40 144)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(73 192)(74 181)(75 182)(76 183)(77 184)(78 185)(79 186)(80 187)(81 188)(82 189)(83 190)(84 191)(97 114)(98 115)(99 116)(100 117)(101 118)(102 119)(103 120)(104 109)(105 110)(106 111)(107 112)(108 113)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 169)

(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)

(1 151 7 145)(2 150 8 156)(3 149 9 155)(4 148 10 154)(5 147 11 153)(6 146 12 152)(13 39 19 45)(14 38 20 44)(15 37 21 43)(16 48 22 42)(17 47 23 41)(18 46 24 40)(25 114 31 120)(26 113 32 119)(27 112 33 118)(28 111 34 117)(29 110 35 116)(30 109 36 115)(49 80 55 74)(50 79 56 73)(51 78 57 84)(52 77 58 83)(53 76 59 82)(54 75 60 81)(61 100 67 106)(62 99 68 105)(63 98 69 104)(64 97 70 103)(65 108 71 102)(66 107 72 101)(85 178 91 172)(86 177 92 171)(87 176 93 170)(88 175 94 169)(89 174 95 180)(90 173 96 179)(121 181 127 187)(122 192 128 186)(123 191 129 185)(124 190 130 184)(125 189 131 183)(126 188 132 182)(133 163 139 157)(134 162 140 168)(135 161 141 167)(136 160 142 166)(137 159 143 165)(138 158 144 164)
 
G:=sub<Sym(192)| (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,73)(23,74)(24,75)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,85)(72,86)(97,173)(98,174)(99,175)(100,176)(101,177)(102,178)(103,179)(104,180)(105,169)(106,170)(107,171)(108,172)(109,155)(110,156)(111,145)(112,146)(113,147)(114,148)(115,149)(116,150)(117,151)(118,152)(119,153)(120,154)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,153)(14,154)(15,155)(16,156)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(73,116)(74,117)(75,118)(76,119)(77,120)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,133)(94,134)(95,135)(96,136)(97,190)(98,191)(99,192)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(157,176)(158,177)(159,178)(160,179)(161,180)(162,169)(163,170)(164,171)(165,172)(166,173)(167,174)(168,175), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,85)(12,86)(13,159)(14,160)(15,161)(16,162)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,157)(24,158)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,61)(35,62)(36,63)(37,141)(38,142)(39,143)(40,144)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(73,192)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(80,187)(81,188)(82,189)(83,190)(84,191)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,109)(105,110)(106,111)(107,112)(108,113)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,169), (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), (1,151,7,145)(2,150,8,156)(3,149,9,155)(4,148,10,154)(5,147,11,153)(6,146,12,152)(13,39,19,45)(14,38,20,44)(15,37,21,43)(16,48,22,42)(17,47,23,41)(18,46,24,40)(25,114,31,120)(26,113,32,119)(27,112,33,118)(28,111,34,117)(29,110,35,116)(30,109,36,115)(49,80,55,74)(50,79,56,73)(51,78,57,84)(52,77,58,83)(53,76,59,82)(54,75,60,81)(61,100,67,106)(62,99,68,105)(63,98,69,104)(64,97,70,103)(65,108,71,102)(66,107,72,101)(85,178,91,172)(86,177,92,171)(87,176,93,170)(88,175,94,169)(89,174,95,180)(90,173,96,179)(121,181,127,187)(122,192,128,186)(123,191,129,185)(124,190,130,184)(125,189,131,183)(126,188,132,182)(133,163,139,157)(134,162,140,168)(135,161,141,167)(136,160,142,166)(137,159,143,165)(138,158,144,164)>;
 
G:=Group( (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,73)(23,74)(24,75)(37,57)(38,58)(39,59)(40,60)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,85)(72,86)(97,173)(98,174)(99,175)(100,176)(101,177)(102,178)(103,179)(104,180)(105,169)(106,170)(107,171)(108,172)(109,155)(110,156)(111,145)(112,146)(113,147)(114,148)(115,149)(116,150)(117,151)(118,152)(119,153)(120,154)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,133)(128,134)(129,135)(130,136)(131,137)(132,138)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192), (1,47)(2,48)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,153)(14,154)(15,155)(16,156)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,58)(26,59)(27,60)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(73,116)(74,117)(75,118)(76,119)(77,120)(78,109)(79,110)(80,111)(81,112)(82,113)(83,114)(84,115)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,133)(94,134)(95,135)(96,136)(97,190)(98,191)(99,192)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(157,176)(158,177)(159,178)(160,179)(161,180)(162,169)(163,170)(164,171)(165,172)(166,173)(167,174)(168,175), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,85)(12,86)(13,159)(14,160)(15,161)(16,162)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,157)(24,158)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,61)(35,62)(36,63)(37,141)(38,142)(39,143)(40,144)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(73,192)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(80,187)(81,188)(82,189)(83,190)(84,191)(97,114)(98,115)(99,116)(100,117)(101,118)(102,119)(103,120)(104,109)(105,110)(106,111)(107,112)(108,113)(145,170)(146,171)(147,172)(148,173)(149,174)(150,175)(151,176)(152,177)(153,178)(154,179)(155,180)(156,169), (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), (1,151,7,145)(2,150,8,156)(3,149,9,155)(4,148,10,154)(5,147,11,153)(6,146,12,152)(13,39,19,45)(14,38,20,44)(15,37,21,43)(16,48,22,42)(17,47,23,41)(18,46,24,40)(25,114,31,120)(26,113,32,119)(27,112,33,118)(28,111,34,117)(29,110,35,116)(30,109,36,115)(49,80,55,74)(50,79,56,73)(51,78,57,84)(52,77,58,83)(53,76,59,82)(54,75,60,81)(61,100,67,106)(62,99,68,105)(63,98,69,104)(64,97,70,103)(65,108,71,102)(66,107,72,101)(85,178,91,172)(86,177,92,171)(87,176,93,170)(88,175,94,169)(89,174,95,180)(90,173,96,179)(121,181,127,187)(122,192,128,186)(123,191,129,185)(124,190,130,184)(125,189,131,183)(126,188,132,182)(133,163,139,157)(134,162,140,168)(135,161,141,167)(136,160,142,166)(137,159,143,165)(138,158,144,164) );
 
G=PermutationGroup([(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,73),(23,74),(24,75),(37,57),(38,58),(39,59),(40,60),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,85),(72,86),(97,173),(98,174),(99,175),(100,176),(101,177),(102,178),(103,179),(104,180),(105,169),(106,170),(107,171),(108,172),(109,155),(110,156),(111,145),(112,146),(113,147),(114,148),(115,149),(116,150),(117,151),(118,152),(119,153),(120,154),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144),(127,133),(128,134),(129,135),(130,136),(131,137),(132,138),(157,181),(158,182),(159,183),(160,184),(161,185),(162,186),(163,187),(164,188),(165,189),(166,190),(167,191),(168,192)], [(1,47),(2,48),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,153),(14,154),(15,155),(16,156),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,58),(26,59),(27,60),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,127),(68,128),(69,129),(70,130),(71,131),(72,132),(73,116),(74,117),(75,118),(76,119),(77,120),(78,109),(79,110),(80,111),(81,112),(82,113),(83,114),(84,115),(85,137),(86,138),(87,139),(88,140),(89,141),(90,142),(91,143),(92,144),(93,133),(94,134),(95,135),(96,136),(97,190),(98,191),(99,192),(100,181),(101,182),(102,183),(103,184),(104,185),(105,186),(106,187),(107,188),(108,189),(157,176),(158,177),(159,178),(160,179),(161,180),(162,169),(163,170),(164,171),(165,172),(166,173),(167,174),(168,175)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,85),(12,86),(13,159),(14,160),(15,161),(16,162),(17,163),(18,164),(19,165),(20,166),(21,167),(22,168),(23,157),(24,158),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,61),(35,62),(36,63),(37,141),(38,142),(39,143),(40,144),(41,133),(42,134),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,127),(50,128),(51,129),(52,130),(53,131),(54,132),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(73,192),(74,181),(75,182),(76,183),(77,184),(78,185),(79,186),(80,187),(81,188),(82,189),(83,190),(84,191),(97,114),(98,115),(99,116),(100,117),(101,118),(102,119),(103,120),(104,109),(105,110),(106,111),(107,112),(108,113),(145,170),(146,171),(147,172),(148,173),(149,174),(150,175),(151,176),(152,177),(153,178),(154,179),(155,180),(156,169)], [(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)], [(1,151,7,145),(2,150,8,156),(3,149,9,155),(4,148,10,154),(5,147,11,153),(6,146,12,152),(13,39,19,45),(14,38,20,44),(15,37,21,43),(16,48,22,42),(17,47,23,41),(18,46,24,40),(25,114,31,120),(26,113,32,119),(27,112,33,118),(28,111,34,117),(29,110,35,116),(30,109,36,115),(49,80,55,74),(50,79,56,73),(51,78,57,84),(52,77,58,83),(53,76,59,82),(54,75,60,81),(61,100,67,106),(62,99,68,105),(63,98,69,104),(64,97,70,103),(65,108,71,102),(66,107,72,101),(85,178,91,172),(86,177,92,171),(87,176,93,170),(88,175,94,169),(89,174,95,180),(90,173,96,179),(121,181,127,187),(122,192,128,186),(123,191,129,185),(124,190,130,184),(125,189,131,183),(126,188,132,182),(133,163,139,157),(134,162,140,168),(135,161,141,167),(136,160,142,166),(137,159,143,165),(138,158,144,164)])
 

Matrix representation ►G ⊆ GL5(𝔽13)

120000
01000
00100
00010
00001
,
120000
012000
001200
00010
00001
,
120000
012000
00100
00010
00001
,
120000
012000
00100
00020
00047
,
120000
01000
001200
00053
00008

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] >;
 

72 conjugacy classes

class 1 2A···2O 3 4A···4H4I···4X6A···6O12A···12P
order12···234···44···46···612···12
size11···122···26···62···22···2

72 irreducible representations

dim111122222
type+++++-++-
imageC1C2C2C2S3Q8D6D6Dic6
kernelC23×Dic6C22×Dic6C23×Dic3C23×C12C23×C4C22×C6C22×C4C24C23
# reps128211814116

In GAP, Magma, Sage, TeX 

C_2^3\times 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
 

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