direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4⋊Dic3, C24.88D6, C23.68D12, C23.21Dic6, C12⋊8(C22×C4), (C22×C12)⋊15C4, C6.42(C23×C4), (C23×C4).17S3, C4⋊2(C22×Dic3), C2.2(C22×D12), C6.31(C22×D4), C6.20(C22×Q8), (C22×C6).28Q8, (C2×C6).283C24, (C23×C12).13C2, (C22×C4)⋊11Dic3, C22.75(C2×D12), (C22×C4).463D6, (C22×C6).148D4, C2.3(C22×Dic6), C2.4(C23×Dic3), (C2×C12).789C23, C22.40(S3×C23), C22.39(C2×Dic6), C23.50(C2×Dic3), C23.344(C22×S3), (C23×C6).105C22, (C22×C6).412C23, (C23×Dic3).10C2, (C22×C12).529C22, (C2×Dic3).277C23, C22.30(C22×Dic3), (C22×Dic3).228C22, C6⋊3(C2×C4⋊C4), C3⋊3(C22×C4⋊C4), (C2×C6)⋊9(C4⋊C4), (C2×C12)⋊36(C2×C4), (C2×C6).54(C2×Q8), (C2×C4)⋊10(C2×Dic3), (C2×C6).181(C2×D4), (C22×C6).142(C2×C4), (C2×C4).739(C22×S3), (C2×C6).207(C22×C4), SmallGroup(192,1344)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 760 in 418 conjugacy classes, 287 normal (15 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×8], C4 [×8], C22, C22 [×34], C6 [×3], C6 [×12], C2×C4 [×28], C2×C4 [×32], C23 [×15], Dic3 [×8], C12 [×8], C2×C6, C2×C6 [×34], C4⋊C4 [×16], C22×C4 [×14], C22×C4 [×20], C24, C2×Dic3 [×8], C2×Dic3 [×24], C2×C12 [×28], C22×C6 [×15], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C4⋊Dic3 [×16], C22×Dic3 [×12], C22×Dic3 [×8], C22×C12 [×14], C23×C6, C22×C4⋊C4, C2×C4⋊Dic3 [×12], C23×Dic3 [×2], C23×C12, C22×C4⋊Dic3
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], Dic3 [×8], D6 [×7], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, Dic6 [×4], D12 [×4], C2×Dic3 [×28], C22×S3 [×7], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C4⋊Dic3 [×16], C2×Dic6 [×6], C2×D12 [×6], C22×Dic3 [×14], S3×C23, C22×C4⋊C4, C2×C4⋊Dic3 [×12], C22×Dic6, C22×D12, C23×Dic3, C22×C4⋊Dic3
Generators and relations
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, ece-1=c-1, ede-1=d-1 >
►Regular action on 192 points
Generators in S192
(1 65)(2 66)(3 61)(4 62)(5 63)(6 64)(7 140)(8 141)(9 142)(10 143)(11 144)(12 139)(13 60)(14 55)(15 56)(16 57)(17 58)(18 59)(19 74)(20 75)(21 76)(22 77)(23 78)(24 73)(25 72)(26 67)(27 68)(28 69)(29 70)(30 71)(31 86)(32 87)(33 88)(34 89)(35 90)(36 85)(37 84)(38 79)(39 80)(40 81)(41 82)(42 83)(43 99)(44 100)(45 101)(46 102)(47 97)(48 98)(49 96)(50 91)(51 92)(52 93)(53 94)(54 95)(103 159)(104 160)(105 161)(106 162)(107 157)(108 158)(109 155)(110 156)(111 151)(112 152)(113 153)(114 154)(115 171)(116 172)(117 173)(118 174)(119 169)(120 170)(121 167)(122 168)(123 163)(124 164)(125 165)(126 166)(127 183)(128 184)(129 185)(130 186)(131 181)(132 182)(133 179)(134 180)(135 175)(136 176)(137 177)(138 178)(145 191)(146 192)(147 187)(148 188)(149 189)(150 190)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 169)(8 170)(9 171)(10 172)(11 173)(12 174)(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(25 52)(26 53)(27 54)(28 49)(29 50)(30 51)(55 82)(56 83)(57 84)(58 79)(59 80)(60 81)(61 88)(62 89)(63 90)(64 85)(65 86)(66 87)(67 94)(68 95)(69 96)(70 91)(71 92)(72 93)(73 101)(74 102)(75 97)(76 98)(77 99)(78 100)(103 130)(104 131)(105 132)(106 127)(107 128)(108 129)(109 136)(110 137)(111 138)(112 133)(113 134)(114 135)(115 142)(116 143)(117 144)(118 139)(119 140)(120 141)(121 148)(122 149)(123 150)(124 145)(125 146)(126 147)(151 178)(152 179)(153 180)(154 175)(155 176)(156 177)(157 184)(158 185)(159 186)(160 181)(161 182)(162 183)(163 190)(164 191)(165 192)(166 187)(167 188)(168 189)
(1 102 17 94)(2 97 18 95)(3 98 13 96)(4 99 14 91)(5 100 15 92)(6 101 16 93)(7 104 188 112)(8 105 189 113)(9 106 190 114)(10 107 191 109)(11 108 192 110)(12 103 187 111)(19 79 26 86)(20 80 27 87)(21 81 28 88)(22 82 29 89)(23 83 30 90)(24 84 25 85)(31 74 38 67)(32 75 39 68)(33 76 40 69)(34 77 41 70)(35 78 42 71)(36 73 37 72)(43 55 50 62)(44 56 51 63)(45 57 52 64)(46 58 53 65)(47 59 54 66)(48 60 49 61)(115 183 123 175)(116 184 124 176)(117 185 125 177)(118 186 126 178)(119 181 121 179)(120 182 122 180)(127 163 135 171)(128 164 136 172)(129 165 137 173)(130 166 138 174)(131 167 133 169)(132 168 134 170)(139 159 147 151)(140 160 148 152)(141 161 149 153)(142 162 150 154)(143 157 145 155)(144 158 146 156)
(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)
(1 119 4 116)(2 118 5 115)(3 117 6 120)(7 89 10 86)(8 88 11 85)(9 87 12 90)(13 125 16 122)(14 124 17 121)(15 123 18 126)(19 112 22 109)(20 111 23 114)(21 110 24 113)(25 105 28 108)(26 104 29 107)(27 103 30 106)(31 140 34 143)(32 139 35 142)(33 144 36 141)(37 149 40 146)(38 148 41 145)(39 147 42 150)(43 136 46 133)(44 135 47 138)(45 134 48 137)(49 129 52 132)(50 128 53 131)(51 127 54 130)(55 164 58 167)(56 163 59 166)(57 168 60 165)(61 173 64 170)(62 172 65 169)(63 171 66 174)(67 160 70 157)(68 159 71 162)(69 158 72 161)(73 153 76 156)(74 152 77 155)(75 151 78 154)(79 188 82 191)(80 187 83 190)(81 192 84 189)(91 184 94 181)(92 183 95 186)(93 182 96 185)(97 178 100 175)(98 177 101 180)(99 176 102 179)
G:=sub<Sym(192)| (1,65)(2,66)(3,61)(4,62)(5,63)(6,64)(7,140)(8,141)(9,142)(10,143)(11,144)(12,139)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,74)(20,75)(21,76)(22,77)(23,78)(24,73)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,86)(32,87)(33,88)(34,89)(35,90)(36,85)(37,84)(38,79)(39,80)(40,81)(41,82)(42,83)(43,99)(44,100)(45,101)(46,102)(47,97)(48,98)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95)(103,159)(104,160)(105,161)(106,162)(107,157)(108,158)(109,155)(110,156)(111,151)(112,152)(113,153)(114,154)(115,171)(116,172)(117,173)(118,174)(119,169)(120,170)(121,167)(122,168)(123,163)(124,164)(125,165)(126,166)(127,183)(128,184)(129,185)(130,186)(131,181)(132,182)(133,179)(134,180)(135,175)(136,176)(137,177)(138,178)(145,191)(146,192)(147,187)(148,188)(149,189)(150,190), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,40)(14,41)(15,42)(16,37)(17,38)(18,39)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,52)(26,53)(27,54)(28,49)(29,50)(30,51)(55,82)(56,83)(57,84)(58,79)(59,80)(60,81)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93)(73,101)(74,102)(75,97)(76,98)(77,99)(78,100)(103,130)(104,131)(105,132)(106,127)(107,128)(108,129)(109,136)(110,137)(111,138)(112,133)(113,134)(114,135)(115,142)(116,143)(117,144)(118,139)(119,140)(120,141)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(151,178)(152,179)(153,180)(154,175)(155,176)(156,177)(157,184)(158,185)(159,186)(160,181)(161,182)(162,183)(163,190)(164,191)(165,192)(166,187)(167,188)(168,189), (1,102,17,94)(2,97,18,95)(3,98,13,96)(4,99,14,91)(5,100,15,92)(6,101,16,93)(7,104,188,112)(8,105,189,113)(9,106,190,114)(10,107,191,109)(11,108,192,110)(12,103,187,111)(19,79,26,86)(20,80,27,87)(21,81,28,88)(22,82,29,89)(23,83,30,90)(24,84,25,85)(31,74,38,67)(32,75,39,68)(33,76,40,69)(34,77,41,70)(35,78,42,71)(36,73,37,72)(43,55,50,62)(44,56,51,63)(45,57,52,64)(46,58,53,65)(47,59,54,66)(48,60,49,61)(115,183,123,175)(116,184,124,176)(117,185,125,177)(118,186,126,178)(119,181,121,179)(120,182,122,180)(127,163,135,171)(128,164,136,172)(129,165,137,173)(130,166,138,174)(131,167,133,169)(132,168,134,170)(139,159,147,151)(140,160,148,152)(141,161,149,153)(142,162,150,154)(143,157,145,155)(144,158,146,156), (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), (1,119,4,116)(2,118,5,115)(3,117,6,120)(7,89,10,86)(8,88,11,85)(9,87,12,90)(13,125,16,122)(14,124,17,121)(15,123,18,126)(19,112,22,109)(20,111,23,114)(21,110,24,113)(25,105,28,108)(26,104,29,107)(27,103,30,106)(31,140,34,143)(32,139,35,142)(33,144,36,141)(37,149,40,146)(38,148,41,145)(39,147,42,150)(43,136,46,133)(44,135,47,138)(45,134,48,137)(49,129,52,132)(50,128,53,131)(51,127,54,130)(55,164,58,167)(56,163,59,166)(57,168,60,165)(61,173,64,170)(62,172,65,169)(63,171,66,174)(67,160,70,157)(68,159,71,162)(69,158,72,161)(73,153,76,156)(74,152,77,155)(75,151,78,154)(79,188,82,191)(80,187,83,190)(81,192,84,189)(91,184,94,181)(92,183,95,186)(93,182,96,185)(97,178,100,175)(98,177,101,180)(99,176,102,179)>;
G:=Group( (1,65)(2,66)(3,61)(4,62)(5,63)(6,64)(7,140)(8,141)(9,142)(10,143)(11,144)(12,139)(13,60)(14,55)(15,56)(16,57)(17,58)(18,59)(19,74)(20,75)(21,76)(22,77)(23,78)(24,73)(25,72)(26,67)(27,68)(28,69)(29,70)(30,71)(31,86)(32,87)(33,88)(34,89)(35,90)(36,85)(37,84)(38,79)(39,80)(40,81)(41,82)(42,83)(43,99)(44,100)(45,101)(46,102)(47,97)(48,98)(49,96)(50,91)(51,92)(52,93)(53,94)(54,95)(103,159)(104,160)(105,161)(106,162)(107,157)(108,158)(109,155)(110,156)(111,151)(112,152)(113,153)(114,154)(115,171)(116,172)(117,173)(118,174)(119,169)(120,170)(121,167)(122,168)(123,163)(124,164)(125,165)(126,166)(127,183)(128,184)(129,185)(130,186)(131,181)(132,182)(133,179)(134,180)(135,175)(136,176)(137,177)(138,178)(145,191)(146,192)(147,187)(148,188)(149,189)(150,190), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,40)(14,41)(15,42)(16,37)(17,38)(18,39)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,52)(26,53)(27,54)(28,49)(29,50)(30,51)(55,82)(56,83)(57,84)(58,79)(59,80)(60,81)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93)(73,101)(74,102)(75,97)(76,98)(77,99)(78,100)(103,130)(104,131)(105,132)(106,127)(107,128)(108,129)(109,136)(110,137)(111,138)(112,133)(113,134)(114,135)(115,142)(116,143)(117,144)(118,139)(119,140)(120,141)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(151,178)(152,179)(153,180)(154,175)(155,176)(156,177)(157,184)(158,185)(159,186)(160,181)(161,182)(162,183)(163,190)(164,191)(165,192)(166,187)(167,188)(168,189), (1,102,17,94)(2,97,18,95)(3,98,13,96)(4,99,14,91)(5,100,15,92)(6,101,16,93)(7,104,188,112)(8,105,189,113)(9,106,190,114)(10,107,191,109)(11,108,192,110)(12,103,187,111)(19,79,26,86)(20,80,27,87)(21,81,28,88)(22,82,29,89)(23,83,30,90)(24,84,25,85)(31,74,38,67)(32,75,39,68)(33,76,40,69)(34,77,41,70)(35,78,42,71)(36,73,37,72)(43,55,50,62)(44,56,51,63)(45,57,52,64)(46,58,53,65)(47,59,54,66)(48,60,49,61)(115,183,123,175)(116,184,124,176)(117,185,125,177)(118,186,126,178)(119,181,121,179)(120,182,122,180)(127,163,135,171)(128,164,136,172)(129,165,137,173)(130,166,138,174)(131,167,133,169)(132,168,134,170)(139,159,147,151)(140,160,148,152)(141,161,149,153)(142,162,150,154)(143,157,145,155)(144,158,146,156), (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), (1,119,4,116)(2,118,5,115)(3,117,6,120)(7,89,10,86)(8,88,11,85)(9,87,12,90)(13,125,16,122)(14,124,17,121)(15,123,18,126)(19,112,22,109)(20,111,23,114)(21,110,24,113)(25,105,28,108)(26,104,29,107)(27,103,30,106)(31,140,34,143)(32,139,35,142)(33,144,36,141)(37,149,40,146)(38,148,41,145)(39,147,42,150)(43,136,46,133)(44,135,47,138)(45,134,48,137)(49,129,52,132)(50,128,53,131)(51,127,54,130)(55,164,58,167)(56,163,59,166)(57,168,60,165)(61,173,64,170)(62,172,65,169)(63,171,66,174)(67,160,70,157)(68,159,71,162)(69,158,72,161)(73,153,76,156)(74,152,77,155)(75,151,78,154)(79,188,82,191)(80,187,83,190)(81,192,84,189)(91,184,94,181)(92,183,95,186)(93,182,96,185)(97,178,100,175)(98,177,101,180)(99,176,102,179) );
G=PermutationGroup([(1,65),(2,66),(3,61),(4,62),(5,63),(6,64),(7,140),(8,141),(9,142),(10,143),(11,144),(12,139),(13,60),(14,55),(15,56),(16,57),(17,58),(18,59),(19,74),(20,75),(21,76),(22,77),(23,78),(24,73),(25,72),(26,67),(27,68),(28,69),(29,70),(30,71),(31,86),(32,87),(33,88),(34,89),(35,90),(36,85),(37,84),(38,79),(39,80),(40,81),(41,82),(42,83),(43,99),(44,100),(45,101),(46,102),(47,97),(48,98),(49,96),(50,91),(51,92),(52,93),(53,94),(54,95),(103,159),(104,160),(105,161),(106,162),(107,157),(108,158),(109,155),(110,156),(111,151),(112,152),(113,153),(114,154),(115,171),(116,172),(117,173),(118,174),(119,169),(120,170),(121,167),(122,168),(123,163),(124,164),(125,165),(126,166),(127,183),(128,184),(129,185),(130,186),(131,181),(132,182),(133,179),(134,180),(135,175),(136,176),(137,177),(138,178),(145,191),(146,192),(147,187),(148,188),(149,189),(150,190)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,169),(8,170),(9,171),(10,172),(11,173),(12,174),(13,40),(14,41),(15,42),(16,37),(17,38),(18,39),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45),(25,52),(26,53),(27,54),(28,49),(29,50),(30,51),(55,82),(56,83),(57,84),(58,79),(59,80),(60,81),(61,88),(62,89),(63,90),(64,85),(65,86),(66,87),(67,94),(68,95),(69,96),(70,91),(71,92),(72,93),(73,101),(74,102),(75,97),(76,98),(77,99),(78,100),(103,130),(104,131),(105,132),(106,127),(107,128),(108,129),(109,136),(110,137),(111,138),(112,133),(113,134),(114,135),(115,142),(116,143),(117,144),(118,139),(119,140),(120,141),(121,148),(122,149),(123,150),(124,145),(125,146),(126,147),(151,178),(152,179),(153,180),(154,175),(155,176),(156,177),(157,184),(158,185),(159,186),(160,181),(161,182),(162,183),(163,190),(164,191),(165,192),(166,187),(167,188),(168,189)], [(1,102,17,94),(2,97,18,95),(3,98,13,96),(4,99,14,91),(5,100,15,92),(6,101,16,93),(7,104,188,112),(8,105,189,113),(9,106,190,114),(10,107,191,109),(11,108,192,110),(12,103,187,111),(19,79,26,86),(20,80,27,87),(21,81,28,88),(22,82,29,89),(23,83,30,90),(24,84,25,85),(31,74,38,67),(32,75,39,68),(33,76,40,69),(34,77,41,70),(35,78,42,71),(36,73,37,72),(43,55,50,62),(44,56,51,63),(45,57,52,64),(46,58,53,65),(47,59,54,66),(48,60,49,61),(115,183,123,175),(116,184,124,176),(117,185,125,177),(118,186,126,178),(119,181,121,179),(120,182,122,180),(127,163,135,171),(128,164,136,172),(129,165,137,173),(130,166,138,174),(131,167,133,169),(132,168,134,170),(139,159,147,151),(140,160,148,152),(141,161,149,153),(142,162,150,154),(143,157,145,155),(144,158,146,156)], [(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)], [(1,119,4,116),(2,118,5,115),(3,117,6,120),(7,89,10,86),(8,88,11,85),(9,87,12,90),(13,125,16,122),(14,124,17,121),(15,123,18,126),(19,112,22,109),(20,111,23,114),(21,110,24,113),(25,105,28,108),(26,104,29,107),(27,103,30,106),(31,140,34,143),(32,139,35,142),(33,144,36,141),(37,149,40,146),(38,148,41,145),(39,147,42,150),(43,136,46,133),(44,135,47,138),(45,134,48,137),(49,129,52,132),(50,128,53,131),(51,127,54,130),(55,164,58,167),(56,163,59,166),(57,168,60,165),(61,173,64,170),(62,172,65,169),(63,171,66,174),(67,160,70,157),(68,159,71,162),(69,158,72,161),(73,153,76,156),(74,152,77,155),(75,151,78,154),(79,188,82,191),(80,187,83,190),(81,192,84,189),(91,184,94,181),(92,183,95,186),(93,182,96,185),(97,178,100,175),(98,177,101,180),(99,176,102,179)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,3,6,0,0,0,0,7,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12] >;
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | D6 | Dic6 | D12 |
| kernel | C22×C4⋊Dic3 | C2×C4⋊Dic3 | C23×Dic3 | C23×C12 | C22×C12 | C23×C4 | C22×C6 | C22×C6 | C22×C4 | C22×C4 | C24 | C23 | C23 |
| # reps | 1 | 12 | 2 | 1 | 16 | 1 | 4 | 4 | 8 | 6 | 1 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4\rtimes Dic_3
% in TeX
G:=Group("C2^2xC4:Dic3"); // GroupNames label
G:=SmallGroup(192,1344);
// by ID
G=gap.SmallGroup(192,1344);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1123,136,6278]);
// 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,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀