direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×C22×C4, C24.86D6, (C2×C6)⋊4C42, C6⋊2(C2×C42), C12⋊9(C22×C4), C3⋊2(C22×C42), (C22×C12)⋊18C4, (C23×C4).21S3, C23.72(C4×S3), C6.36(C23×C4), (C2×C6).280C24, (C23×C12).21C2, (C22×C4).489D6, C2.2(C23×Dic3), (C2×C12).884C23, C22.38(S3×C23), C23.49(C2×Dic3), (C22×C6).409C23, C23.342(C22×S3), (C23×C6).102C22, (C23×Dic3).12C2, (C22×C12).569C22, (C2×Dic3).314C23, C22.29(C22×Dic3), (C22×Dic3).244C22, (C2×C12)⋊38(C2×C4), C2.3(S3×C22×C4), C22.77(S3×C2×C4), (C2×C4).828(C22×S3), (C22×C6).141(C2×C4), (C2×C6).206(C22×C4), SmallGroup(192,1341)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C22×C4 |
Generators and relations for Dic3×C22×C4
G = < a,b,c,d,e | a2=b2=c4=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 760 in 498 conjugacy classes, 367 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C22×C6, C2×C42, C23×C4, C23×C4, C4×Dic3, C22×Dic3, C22×C12, C23×C6, C22×C42, C2×C4×Dic3, C23×Dic3, C23×C12, Dic3×C22×C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, C24, C4×S3, C2×Dic3, C22×S3, C2×C42, C23×C4, C4×Dic3, S3×C2×C4, C22×Dic3, S3×C23, C22×C42, C2×C4×Dic3, S3×C22×C4, C23×Dic3, Dic3×C22×C4
►Regular action on 192 points
Generators in S192
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 142)(8 143)(9 144)(10 139)(11 140)(12 141)(13 56)(14 57)(15 58)(16 59)(17 60)(18 55)(19 78)(20 73)(21 74)(22 75)(23 76)(24 77)(25 68)(26 69)(27 70)(28 71)(29 72)(30 67)(31 90)(32 85)(33 86)(34 87)(35 88)(36 89)(37 80)(38 81)(39 82)(40 83)(41 84)(42 79)(43 101)(44 102)(45 97)(46 98)(47 99)(48 100)(49 92)(50 93)(51 94)(52 95)(53 96)(54 91)(103 161)(104 162)(105 157)(106 158)(107 159)(108 160)(109 153)(110 154)(111 155)(112 156)(113 151)(114 152)(115 173)(116 174)(117 169)(118 170)(119 171)(120 172)(121 165)(122 166)(123 167)(124 168)(125 163)(126 164)(127 185)(128 186)(129 181)(130 182)(131 183)(132 184)(133 177)(134 178)(135 179)(136 180)(137 175)(138 176)(145 189)(146 190)(147 191)(148 192)(149 187)(150 188)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 166)(8 167)(9 168)(10 163)(11 164)(12 165)(13 32)(14 33)(15 34)(16 35)(17 36)(18 31)(19 54)(20 49)(21 50)(22 51)(23 52)(24 53)(25 44)(26 45)(27 46)(28 47)(29 48)(30 43)(55 90)(56 85)(57 86)(58 87)(59 88)(60 89)(61 80)(62 81)(63 82)(64 83)(65 84)(66 79)(67 101)(68 102)(69 97)(70 98)(71 99)(72 100)(73 92)(74 93)(75 94)(76 95)(77 96)(78 91)(103 137)(104 138)(105 133)(106 134)(107 135)(108 136)(109 129)(110 130)(111 131)(112 132)(113 127)(114 128)(115 149)(116 150)(117 145)(118 146)(119 147)(120 148)(121 141)(122 142)(123 143)(124 144)(125 139)(126 140)(151 185)(152 186)(153 181)(154 182)(155 183)(156 184)(157 177)(158 178)(159 179)(160 180)(161 175)(162 176)(169 189)(170 190)(171 191)(172 192)(173 187)(174 188)
(1 28 16 20)(2 29 17 21)(3 30 18 22)(4 25 13 23)(5 26 14 24)(6 27 15 19)(7 182 187 175)(8 183 188 176)(9 184 189 177)(10 185 190 178)(11 186 191 179)(12 181 192 180)(31 51 39 43)(32 52 40 44)(33 53 41 45)(34 54 42 46)(35 49 37 47)(36 50 38 48)(55 75 63 67)(56 76 64 68)(57 77 65 69)(58 78 66 70)(59 73 61 71)(60 74 62 72)(79 98 87 91)(80 99 88 92)(81 100 89 93)(82 101 90 94)(83 102 85 95)(84 97 86 96)(103 122 110 115)(104 123 111 116)(105 124 112 117)(106 125 113 118)(107 126 114 119)(108 121 109 120)(127 146 134 139)(128 147 135 140)(129 148 136 141)(130 149 137 142)(131 150 138 143)(132 145 133 144)(151 170 158 163)(152 171 159 164)(153 172 160 165)(154 173 161 166)(155 174 162 167)(156 169 157 168)
(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)
(1 175 4 178)(2 180 5 177)(3 179 6 176)(7 25 10 28)(8 30 11 27)(9 29 12 26)(13 185 16 182)(14 184 17 181)(15 183 18 186)(19 188 22 191)(20 187 23 190)(21 192 24 189)(31 152 34 155)(32 151 35 154)(33 156 36 153)(37 161 40 158)(38 160 41 157)(39 159 42 162)(43 164 46 167)(44 163 47 166)(45 168 48 165)(49 173 52 170)(50 172 53 169)(51 171 54 174)(55 128 58 131)(56 127 59 130)(57 132 60 129)(61 137 64 134)(62 136 65 133)(63 135 66 138)(67 140 70 143)(68 139 71 142)(69 144 72 141)(73 149 76 146)(74 148 77 145)(75 147 78 150)(79 104 82 107)(80 103 83 106)(81 108 84 105)(85 113 88 110)(86 112 89 109)(87 111 90 114)(91 116 94 119)(92 115 95 118)(93 120 96 117)(97 124 100 121)(98 123 101 126)(99 122 102 125)
G:=sub<Sym(192)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,142)(8,143)(9,144)(10,139)(11,140)(12,141)(13,56)(14,57)(15,58)(16,59)(17,60)(18,55)(19,78)(20,73)(21,74)(22,75)(23,76)(24,77)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,90)(32,85)(33,86)(34,87)(35,88)(36,89)(37,80)(38,81)(39,82)(40,83)(41,84)(42,79)(43,101)(44,102)(45,97)(46,98)(47,99)(48,100)(49,92)(50,93)(51,94)(52,95)(53,96)(54,91)(103,161)(104,162)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)(113,151)(114,152)(115,173)(116,174)(117,169)(118,170)(119,171)(120,172)(121,165)(122,166)(123,167)(124,168)(125,163)(126,164)(127,185)(128,186)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)(137,175)(138,176)(145,189)(146,190)(147,191)(148,192)(149,187)(150,188), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,166)(8,167)(9,168)(10,163)(11,164)(12,165)(13,32)(14,33)(15,34)(16,35)(17,36)(18,31)(19,54)(20,49)(21,50)(22,51)(23,52)(24,53)(25,44)(26,45)(27,46)(28,47)(29,48)(30,43)(55,90)(56,85)(57,86)(58,87)(59,88)(60,89)(61,80)(62,81)(63,82)(64,83)(65,84)(66,79)(67,101)(68,102)(69,97)(70,98)(71,99)(72,100)(73,92)(74,93)(75,94)(76,95)(77,96)(78,91)(103,137)(104,138)(105,133)(106,134)(107,135)(108,136)(109,129)(110,130)(111,131)(112,132)(113,127)(114,128)(115,149)(116,150)(117,145)(118,146)(119,147)(120,148)(121,141)(122,142)(123,143)(124,144)(125,139)(126,140)(151,185)(152,186)(153,181)(154,182)(155,183)(156,184)(157,177)(158,178)(159,179)(160,180)(161,175)(162,176)(169,189)(170,190)(171,191)(172,192)(173,187)(174,188), (1,28,16,20)(2,29,17,21)(3,30,18,22)(4,25,13,23)(5,26,14,24)(6,27,15,19)(7,182,187,175)(8,183,188,176)(9,184,189,177)(10,185,190,178)(11,186,191,179)(12,181,192,180)(31,51,39,43)(32,52,40,44)(33,53,41,45)(34,54,42,46)(35,49,37,47)(36,50,38,48)(55,75,63,67)(56,76,64,68)(57,77,65,69)(58,78,66,70)(59,73,61,71)(60,74,62,72)(79,98,87,91)(80,99,88,92)(81,100,89,93)(82,101,90,94)(83,102,85,95)(84,97,86,96)(103,122,110,115)(104,123,111,116)(105,124,112,117)(106,125,113,118)(107,126,114,119)(108,121,109,120)(127,146,134,139)(128,147,135,140)(129,148,136,141)(130,149,137,142)(131,150,138,143)(132,145,133,144)(151,170,158,163)(152,171,159,164)(153,172,160,165)(154,173,161,166)(155,174,162,167)(156,169,157,168), (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), (1,175,4,178)(2,180,5,177)(3,179,6,176)(7,25,10,28)(8,30,11,27)(9,29,12,26)(13,185,16,182)(14,184,17,181)(15,183,18,186)(19,188,22,191)(20,187,23,190)(21,192,24,189)(31,152,34,155)(32,151,35,154)(33,156,36,153)(37,161,40,158)(38,160,41,157)(39,159,42,162)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,173,52,170)(50,172,53,169)(51,171,54,174)(55,128,58,131)(56,127,59,130)(57,132,60,129)(61,137,64,134)(62,136,65,133)(63,135,66,138)(67,140,70,143)(68,139,71,142)(69,144,72,141)(73,149,76,146)(74,148,77,145)(75,147,78,150)(79,104,82,107)(80,103,83,106)(81,108,84,105)(85,113,88,110)(86,112,89,109)(87,111,90,114)(91,116,94,119)(92,115,95,118)(93,120,96,117)(97,124,100,121)(98,123,101,126)(99,122,102,125)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,142)(8,143)(9,144)(10,139)(11,140)(12,141)(13,56)(14,57)(15,58)(16,59)(17,60)(18,55)(19,78)(20,73)(21,74)(22,75)(23,76)(24,77)(25,68)(26,69)(27,70)(28,71)(29,72)(30,67)(31,90)(32,85)(33,86)(34,87)(35,88)(36,89)(37,80)(38,81)(39,82)(40,83)(41,84)(42,79)(43,101)(44,102)(45,97)(46,98)(47,99)(48,100)(49,92)(50,93)(51,94)(52,95)(53,96)(54,91)(103,161)(104,162)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)(113,151)(114,152)(115,173)(116,174)(117,169)(118,170)(119,171)(120,172)(121,165)(122,166)(123,167)(124,168)(125,163)(126,164)(127,185)(128,186)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)(137,175)(138,176)(145,189)(146,190)(147,191)(148,192)(149,187)(150,188), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,166)(8,167)(9,168)(10,163)(11,164)(12,165)(13,32)(14,33)(15,34)(16,35)(17,36)(18,31)(19,54)(20,49)(21,50)(22,51)(23,52)(24,53)(25,44)(26,45)(27,46)(28,47)(29,48)(30,43)(55,90)(56,85)(57,86)(58,87)(59,88)(60,89)(61,80)(62,81)(63,82)(64,83)(65,84)(66,79)(67,101)(68,102)(69,97)(70,98)(71,99)(72,100)(73,92)(74,93)(75,94)(76,95)(77,96)(78,91)(103,137)(104,138)(105,133)(106,134)(107,135)(108,136)(109,129)(110,130)(111,131)(112,132)(113,127)(114,128)(115,149)(116,150)(117,145)(118,146)(119,147)(120,148)(121,141)(122,142)(123,143)(124,144)(125,139)(126,140)(151,185)(152,186)(153,181)(154,182)(155,183)(156,184)(157,177)(158,178)(159,179)(160,180)(161,175)(162,176)(169,189)(170,190)(171,191)(172,192)(173,187)(174,188), (1,28,16,20)(2,29,17,21)(3,30,18,22)(4,25,13,23)(5,26,14,24)(6,27,15,19)(7,182,187,175)(8,183,188,176)(9,184,189,177)(10,185,190,178)(11,186,191,179)(12,181,192,180)(31,51,39,43)(32,52,40,44)(33,53,41,45)(34,54,42,46)(35,49,37,47)(36,50,38,48)(55,75,63,67)(56,76,64,68)(57,77,65,69)(58,78,66,70)(59,73,61,71)(60,74,62,72)(79,98,87,91)(80,99,88,92)(81,100,89,93)(82,101,90,94)(83,102,85,95)(84,97,86,96)(103,122,110,115)(104,123,111,116)(105,124,112,117)(106,125,113,118)(107,126,114,119)(108,121,109,120)(127,146,134,139)(128,147,135,140)(129,148,136,141)(130,149,137,142)(131,150,138,143)(132,145,133,144)(151,170,158,163)(152,171,159,164)(153,172,160,165)(154,173,161,166)(155,174,162,167)(156,169,157,168), (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), (1,175,4,178)(2,180,5,177)(3,179,6,176)(7,25,10,28)(8,30,11,27)(9,29,12,26)(13,185,16,182)(14,184,17,181)(15,183,18,186)(19,188,22,191)(20,187,23,190)(21,192,24,189)(31,152,34,155)(32,151,35,154)(33,156,36,153)(37,161,40,158)(38,160,41,157)(39,159,42,162)(43,164,46,167)(44,163,47,166)(45,168,48,165)(49,173,52,170)(50,172,53,169)(51,171,54,174)(55,128,58,131)(56,127,59,130)(57,132,60,129)(61,137,64,134)(62,136,65,133)(63,135,66,138)(67,140,70,143)(68,139,71,142)(69,144,72,141)(73,149,76,146)(74,148,77,145)(75,147,78,150)(79,104,82,107)(80,103,83,106)(81,108,84,105)(85,113,88,110)(86,112,89,109)(87,111,90,114)(91,116,94,119)(92,115,95,118)(93,120,96,117)(97,124,100,121)(98,123,101,126)(99,122,102,125) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,142),(8,143),(9,144),(10,139),(11,140),(12,141),(13,56),(14,57),(15,58),(16,59),(17,60),(18,55),(19,78),(20,73),(21,74),(22,75),(23,76),(24,77),(25,68),(26,69),(27,70),(28,71),(29,72),(30,67),(31,90),(32,85),(33,86),(34,87),(35,88),(36,89),(37,80),(38,81),(39,82),(40,83),(41,84),(42,79),(43,101),(44,102),(45,97),(46,98),(47,99),(48,100),(49,92),(50,93),(51,94),(52,95),(53,96),(54,91),(103,161),(104,162),(105,157),(106,158),(107,159),(108,160),(109,153),(110,154),(111,155),(112,156),(113,151),(114,152),(115,173),(116,174),(117,169),(118,170),(119,171),(120,172),(121,165),(122,166),(123,167),(124,168),(125,163),(126,164),(127,185),(128,186),(129,181),(130,182),(131,183),(132,184),(133,177),(134,178),(135,179),(136,180),(137,175),(138,176),(145,189),(146,190),(147,191),(148,192),(149,187),(150,188)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,166),(8,167),(9,168),(10,163),(11,164),(12,165),(13,32),(14,33),(15,34),(16,35),(17,36),(18,31),(19,54),(20,49),(21,50),(22,51),(23,52),(24,53),(25,44),(26,45),(27,46),(28,47),(29,48),(30,43),(55,90),(56,85),(57,86),(58,87),(59,88),(60,89),(61,80),(62,81),(63,82),(64,83),(65,84),(66,79),(67,101),(68,102),(69,97),(70,98),(71,99),(72,100),(73,92),(74,93),(75,94),(76,95),(77,96),(78,91),(103,137),(104,138),(105,133),(106,134),(107,135),(108,136),(109,129),(110,130),(111,131),(112,132),(113,127),(114,128),(115,149),(116,150),(117,145),(118,146),(119,147),(120,148),(121,141),(122,142),(123,143),(124,144),(125,139),(126,140),(151,185),(152,186),(153,181),(154,182),(155,183),(156,184),(157,177),(158,178),(159,179),(160,180),(161,175),(162,176),(169,189),(170,190),(171,191),(172,192),(173,187),(174,188)], [(1,28,16,20),(2,29,17,21),(3,30,18,22),(4,25,13,23),(5,26,14,24),(6,27,15,19),(7,182,187,175),(8,183,188,176),(9,184,189,177),(10,185,190,178),(11,186,191,179),(12,181,192,180),(31,51,39,43),(32,52,40,44),(33,53,41,45),(34,54,42,46),(35,49,37,47),(36,50,38,48),(55,75,63,67),(56,76,64,68),(57,77,65,69),(58,78,66,70),(59,73,61,71),(60,74,62,72),(79,98,87,91),(80,99,88,92),(81,100,89,93),(82,101,90,94),(83,102,85,95),(84,97,86,96),(103,122,110,115),(104,123,111,116),(105,124,112,117),(106,125,113,118),(107,126,114,119),(108,121,109,120),(127,146,134,139),(128,147,135,140),(129,148,136,141),(130,149,137,142),(131,150,138,143),(132,145,133,144),(151,170,158,163),(152,171,159,164),(153,172,160,165),(154,173,161,166),(155,174,162,167),(156,169,157,168)], [(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)], [(1,175,4,178),(2,180,5,177),(3,179,6,176),(7,25,10,28),(8,30,11,27),(9,29,12,26),(13,185,16,182),(14,184,17,181),(15,183,18,186),(19,188,22,191),(20,187,23,190),(21,192,24,189),(31,152,34,155),(32,151,35,154),(33,156,36,153),(37,161,40,158),(38,160,41,157),(39,159,42,162),(43,164,46,167),(44,163,47,166),(45,168,48,165),(49,173,52,170),(50,172,53,169),(51,171,54,174),(55,128,58,131),(56,127,59,130),(57,132,60,129),(61,137,64,134),(62,136,65,133),(63,135,66,138),(67,140,70,143),(68,139,71,142),(69,144,72,141),(73,149,76,146),(74,148,77,145),(75,147,78,150),(79,104,82,107),(80,103,83,106),(81,108,84,105),(85,113,88,110),(86,112,89,109),(87,111,90,114),(91,116,94,119),(92,115,95,118),(93,120,96,117),(97,124,100,121),(98,123,101,126),(99,122,102,125)]])
96 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3 | 4A | ··· | 4P | 4Q | ··· | 4AV | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | D6 | C4×S3 |
| kernel | Dic3×C22×C4 | C2×C4×Dic3 | C23×Dic3 | C23×C12 | C22×Dic3 | C22×C12 | C23×C4 | C22×C4 | C22×C4 | C24 | C23 |
| # reps | 1 | 12 | 2 | 1 | 32 | 16 | 1 | 8 | 6 | 1 | 16 |
Matrix representation of Dic3×C22×C4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 7 | 3 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,1,0],[8,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,0,10,7,0,0,0,10,3] >;
Dic3×C22×C4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_2^2\times C_4 % in TeX
G:=Group("Dic3xC2^2xC4"); // GroupNames label
G:=SmallGroup(192,1341);
// by ID
G=gap.SmallGroup(192,1341);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^6=1,e^2=d^3,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