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## G = Dic3×C24order 192 = 26·3

### Direct product of C24 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C24
 Chief series C1 — C3 — C6 — Dic3 — C2×Dic3 — C22×Dic3 — C23×Dic3 — Dic3×C24
 Lower central C3 — Dic3×C24
 Upper central C1 — C25

Generators and relations for Dic3×C24
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=e3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1976 in 1362 conjugacy classes, 1055 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C2×C4, C23, Dic3, C2×C6, C22×C4, C24, C2×Dic3, C22×C6, C23×C4, C25, C22×Dic3, C23×C6, C24×C4, C23×Dic3, C24×C6, Dic3×C24
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C24, C2×Dic3, C22×S3, C23×C4, C25, C22×Dic3, S3×C23, C24×C4, C23×Dic3, S3×C24, Dic3×C24

Smallest permutation representation of Dic3×C24
Regular action on 192 points
Generators in S192
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 58)(8 59)(9 60)(10 55)(11 56)(12 57)(13 64)(14 65)(15 66)(16 61)(17 62)(18 63)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 76)(26 77)(27 78)(28 73)(29 74)(30 75)(31 82)(32 83)(33 84)(34 79)(35 80)(36 81)(37 88)(38 89)(39 90)(40 85)(41 86)(42 87)(43 94)(44 95)(45 96)(46 91)(47 92)(48 93)(97 148)(98 149)(99 150)(100 145)(101 146)(102 147)(103 154)(104 155)(105 156)(106 151)(107 152)(108 153)(109 160)(110 161)(111 162)(112 157)(113 158)(114 159)(115 166)(116 167)(117 168)(118 163)(119 164)(120 165)(121 172)(122 173)(123 174)(124 169)(125 170)(126 171)(127 178)(128 179)(129 180)(130 175)(131 176)(132 177)(133 184)(134 185)(135 186)(136 181)(137 182)(138 183)(139 190)(140 191)(141 192)(142 187)(143 188)(144 189)
(1 47)(2 48)(3 43)(4 44)(5 45)(6 46)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)(157 175)(158 176)(159 177)(160 178)(161 179)(162 180)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 109)(104 110)(105 111)(106 112)(107 113)(108 114)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)(127 133)(128 134)(129 135)(130 136)(131 137)(132 138)(145 163)(146 164)(147 165)(148 166)(149 167)(150 168)(151 157)(152 158)(153 159)(154 160)(155 161)(156 162)(169 187)(170 188)(171 189)(172 190)(173 191)(174 192)(175 181)(176 182)(177 183)(178 184)(179 185)(180 186)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)(49 58)(50 59)(51 60)(52 55)(53 56)(54 57)(61 70)(62 71)(63 72)(64 67)(65 68)(66 69)(73 82)(74 83)(75 84)(76 79)(77 80)(78 81)(85 94)(86 95)(87 96)(88 91)(89 92)(90 93)(97 106)(98 107)(99 108)(100 103)(101 104)(102 105)(109 118)(110 119)(111 120)(112 115)(113 116)(114 117)(121 130)(122 131)(123 132)(124 127)(125 128)(126 129)(133 142)(134 143)(135 144)(136 139)(137 140)(138 141)(145 154)(146 155)(147 156)(148 151)(149 152)(150 153)(157 166)(158 167)(159 168)(160 163)(161 164)(162 165)(169 178)(170 179)(171 180)(172 175)(173 176)(174 177)(181 190)(182 191)(183 192)(184 187)(185 188)(186 189)
(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)
(1 124 4 121)(2 123 5 126)(3 122 6 125)(7 128 10 131)(8 127 11 130)(9 132 12 129)(13 134 16 137)(14 133 17 136)(15 138 18 135)(19 140 22 143)(20 139 23 142)(21 144 24 141)(25 98 28 101)(26 97 29 100)(27 102 30 99)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 110 40 113)(38 109 41 112)(39 114 42 111)(43 116 46 119)(44 115 47 118)(45 120 48 117)(49 170 52 173)(50 169 53 172)(51 174 54 171)(55 176 58 179)(56 175 59 178)(57 180 60 177)(61 182 64 185)(62 181 65 184)(63 186 66 183)(67 188 70 191)(68 187 71 190)(69 192 72 189)(73 146 76 149)(74 145 77 148)(75 150 78 147)(79 152 82 155)(80 151 83 154)(81 156 84 153)(85 158 88 161)(86 157 89 160)(87 162 90 159)(91 164 94 167)(92 163 95 166)(93 168 96 165)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2AE 3 4A ··· 4AF 6A ··· 6AE order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 size 1 1 ··· 1 2 3 ··· 3 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 S3 Dic3 D6 kernel Dic3×C24 C23×Dic3 C24×C6 C23×C6 C25 C24 C24 # reps 1 30 1 32 1 16 15

Matrix representation of Dic3×C24 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 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
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 12 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

Dic3×C24 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2^4
% in TeX

G:=Group("Dic3xC2^4");
// GroupNames label

G:=SmallGroup(192,1528);
// by ID

G=gap.SmallGroup(192,1528);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=e^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽