direct product, metabelian, supersoluble, monomial
Aliases: C2×C33⋊6D4, C62.113D6, (S3×C6)⋊18D6, (C32×C6)⋊6D4, C33⋊21(C2×D4), (S3×C62)⋊4C2, C6⋊2(D6⋊S3), C6⋊2(C32⋊7D4), C33⋊5C4⋊11C22, (C3×C62).29C22, (C32×C6).58C23, (S3×C2×C6)⋊6S3, C6.68(C2×S32), D6⋊5(C2×C3⋊S3), (C2×C6).42S32, (C2×C3⋊S3)⋊19D6, C3⋊3(C2×D6⋊S3), (C3×C6)⋊6(C3⋊D4), (C22×C3⋊S3)⋊8S3, (S3×C3×C6)⋊18C22, C3⋊3(C2×C32⋊7D4), (C6×C3⋊S3)⋊16C22, (C2×C33⋊5C4)⋊8C2, C22.14(S3×C3⋊S3), C6.21(C22×C3⋊S3), C32⋊11(C2×C3⋊D4), (C22×S3)⋊2(C3⋊S3), (C3×C6).147(C22×S3), (C2×C6×C3⋊S3)⋊3C2, C2.21(C2×S3×C3⋊S3), (C2×C6).23(C2×C3⋊S3), SmallGroup(432,680)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C33⋊6D4
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=b-1, bf=fb, cd=dc, ece-1=c-1, cf=fc, ede-1=fdf=d-1, fef=e-1 >
Subgroups: 1880 in 388 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×S3, C22×C6, C33, C3⋊Dic3, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×C3⋊D4, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, D6⋊S3, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, S3×C2×C6, C22×C3⋊S3, C2×C62, C33⋊5C4, S3×C3×C6, S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3, C3×C62, C2×D6⋊S3, C2×C32⋊7D4, C33⋊6D4, C2×C33⋊5C4, S3×C62, C2×C6×C3⋊S3, C2×C33⋊6D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C3⋊D4, C22×S3, S32, C2×C3⋊S3, C2×C3⋊D4, D6⋊S3, C32⋊7D4, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C2×D6⋊S3, C2×C32⋊7D4, C33⋊6D4, C2×S3×C3⋊S3, C2×C33⋊6D4
►On 144 points
Generators in S144
(1 112)(2 109)(3 110)(4 111)(5 61)(6 62)(7 63)(8 64)(9 42)(10 43)(11 44)(12 41)(13 90)(14 91)(15 92)(16 89)(17 68)(18 65)(19 66)(20 67)(21 102)(22 103)(23 104)(24 101)(25 50)(26 51)(27 52)(28 49)(29 87)(30 88)(31 85)(32 86)(33 47)(34 48)(35 45)(36 46)(37 118)(38 119)(39 120)(40 117)(53 130)(54 131)(55 132)(56 129)(57 125)(58 126)(59 127)(60 128)(69 107)(70 108)(71 105)(72 106)(73 134)(74 135)(75 136)(76 133)(77 138)(78 139)(79 140)(80 137)(81 143)(82 144)(83 141)(84 142)(93 124)(94 121)(95 122)(96 123)(97 113)(98 114)(99 115)(100 116)
(1 40 42)(2 43 37)(3 38 44)(4 41 39)(5 106 115)(6 116 107)(7 108 113)(8 114 105)(9 112 117)(10 118 109)(11 110 119)(12 120 111)(13 96 65)(14 66 93)(15 94 67)(16 68 95)(17 122 89)(18 90 123)(19 124 91)(20 92 121)(21 125 129)(22 130 126)(23 127 131)(24 132 128)(25 32 140)(26 137 29)(27 30 138)(28 139 31)(33 73 141)(34 142 74)(35 75 143)(36 144 76)(45 136 81)(46 82 133)(47 134 83)(48 84 135)(49 78 85)(50 86 79)(51 80 87)(52 88 77)(53 58 103)(54 104 59)(55 60 101)(56 102 57)(61 72 99)(62 100 69)(63 70 97)(64 98 71)
(1 134 29)(2 30 135)(3 136 31)(4 32 133)(5 104 95)(6 96 101)(7 102 93)(8 94 103)(9 33 80)(10 77 34)(11 35 78)(12 79 36)(13 60 107)(14 108 57)(15 58 105)(16 106 59)(17 99 131)(18 132 100)(19 97 129)(20 130 98)(21 124 63)(22 64 121)(23 122 61)(24 62 123)(25 82 39)(26 40 83)(27 84 37)(28 38 81)(41 140 46)(42 47 137)(43 138 48)(44 45 139)(49 119 143)(50 144 120)(51 117 141)(52 142 118)(53 114 67)(54 68 115)(55 116 65)(56 66 113)(69 90 128)(70 125 91)(71 92 126)(72 127 89)(73 87 112)(74 109 88)(75 85 110)(76 111 86)
(1 26 47)(2 48 27)(3 28 45)(4 46 25)(5 54 16)(6 13 55)(7 56 14)(8 15 53)(9 87 141)(10 142 88)(11 85 143)(12 144 86)(17 72 23)(18 24 69)(19 70 21)(20 22 71)(29 83 42)(30 43 84)(31 81 44)(32 41 82)(33 112 51)(34 52 109)(35 110 49)(36 50 111)(37 135 138)(38 139 136)(39 133 140)(40 137 134)(57 93 113)(58 114 94)(59 95 115)(60 116 96)(61 131 89)(62 90 132)(63 129 91)(64 92 130)(65 101 107)(66 108 102)(67 103 105)(68 106 104)(73 117 80)(74 77 118)(75 119 78)(76 79 120)(97 125 124)(98 121 126)(99 127 122)(100 123 128)
(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)
(1 57)(2 60)(3 59)(4 58)(5 139)(6 138)(7 137)(8 140)(9 21)(10 24)(11 23)(12 22)(13 135)(14 134)(15 133)(16 136)(17 143)(18 142)(19 141)(20 144)(25 114)(26 113)(27 116)(28 115)(29 108)(30 107)(31 106)(32 105)(33 124)(34 123)(35 122)(36 121)(37 55)(38 54)(39 53)(40 56)(41 103)(42 102)(43 101)(44 104)(45 95)(46 94)(47 93)(48 96)(49 99)(50 98)(51 97)(52 100)(61 78)(62 77)(63 80)(64 79)(65 84)(66 83)(67 82)(68 81)(69 88)(70 87)(71 86)(72 85)(73 91)(74 90)(75 89)(76 92)(109 128)(110 127)(111 126)(112 125)(117 129)(118 132)(119 131)(120 130)
G:=sub<Sym(144)| (1,112)(2,109)(3,110)(4,111)(5,61)(6,62)(7,63)(8,64)(9,42)(10,43)(11,44)(12,41)(13,90)(14,91)(15,92)(16,89)(17,68)(18,65)(19,66)(20,67)(21,102)(22,103)(23,104)(24,101)(25,50)(26,51)(27,52)(28,49)(29,87)(30,88)(31,85)(32,86)(33,47)(34,48)(35,45)(36,46)(37,118)(38,119)(39,120)(40,117)(53,130)(54,131)(55,132)(56,129)(57,125)(58,126)(59,127)(60,128)(69,107)(70,108)(71,105)(72,106)(73,134)(74,135)(75,136)(76,133)(77,138)(78,139)(79,140)(80,137)(81,143)(82,144)(83,141)(84,142)(93,124)(94,121)(95,122)(96,123)(97,113)(98,114)(99,115)(100,116), (1,40,42)(2,43,37)(3,38,44)(4,41,39)(5,106,115)(6,116,107)(7,108,113)(8,114,105)(9,112,117)(10,118,109)(11,110,119)(12,120,111)(13,96,65)(14,66,93)(15,94,67)(16,68,95)(17,122,89)(18,90,123)(19,124,91)(20,92,121)(21,125,129)(22,130,126)(23,127,131)(24,132,128)(25,32,140)(26,137,29)(27,30,138)(28,139,31)(33,73,141)(34,142,74)(35,75,143)(36,144,76)(45,136,81)(46,82,133)(47,134,83)(48,84,135)(49,78,85)(50,86,79)(51,80,87)(52,88,77)(53,58,103)(54,104,59)(55,60,101)(56,102,57)(61,72,99)(62,100,69)(63,70,97)(64,98,71), (1,134,29)(2,30,135)(3,136,31)(4,32,133)(5,104,95)(6,96,101)(7,102,93)(8,94,103)(9,33,80)(10,77,34)(11,35,78)(12,79,36)(13,60,107)(14,108,57)(15,58,105)(16,106,59)(17,99,131)(18,132,100)(19,97,129)(20,130,98)(21,124,63)(22,64,121)(23,122,61)(24,62,123)(25,82,39)(26,40,83)(27,84,37)(28,38,81)(41,140,46)(42,47,137)(43,138,48)(44,45,139)(49,119,143)(50,144,120)(51,117,141)(52,142,118)(53,114,67)(54,68,115)(55,116,65)(56,66,113)(69,90,128)(70,125,91)(71,92,126)(72,127,89)(73,87,112)(74,109,88)(75,85,110)(76,111,86), (1,26,47)(2,48,27)(3,28,45)(4,46,25)(5,54,16)(6,13,55)(7,56,14)(8,15,53)(9,87,141)(10,142,88)(11,85,143)(12,144,86)(17,72,23)(18,24,69)(19,70,21)(20,22,71)(29,83,42)(30,43,84)(31,81,44)(32,41,82)(33,112,51)(34,52,109)(35,110,49)(36,50,111)(37,135,138)(38,139,136)(39,133,140)(40,137,134)(57,93,113)(58,114,94)(59,95,115)(60,116,96)(61,131,89)(62,90,132)(63,129,91)(64,92,130)(65,101,107)(66,108,102)(67,103,105)(68,106,104)(73,117,80)(74,77,118)(75,119,78)(76,79,120)(97,125,124)(98,121,126)(99,127,122)(100,123,128), (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), (1,57)(2,60)(3,59)(4,58)(5,139)(6,138)(7,137)(8,140)(9,21)(10,24)(11,23)(12,22)(13,135)(14,134)(15,133)(16,136)(17,143)(18,142)(19,141)(20,144)(25,114)(26,113)(27,116)(28,115)(29,108)(30,107)(31,106)(32,105)(33,124)(34,123)(35,122)(36,121)(37,55)(38,54)(39,53)(40,56)(41,103)(42,102)(43,101)(44,104)(45,95)(46,94)(47,93)(48,96)(49,99)(50,98)(51,97)(52,100)(61,78)(62,77)(63,80)(64,79)(65,84)(66,83)(67,82)(68,81)(69,88)(70,87)(71,86)(72,85)(73,91)(74,90)(75,89)(76,92)(109,128)(110,127)(111,126)(112,125)(117,129)(118,132)(119,131)(120,130)>;
G:=Group( (1,112)(2,109)(3,110)(4,111)(5,61)(6,62)(7,63)(8,64)(9,42)(10,43)(11,44)(12,41)(13,90)(14,91)(15,92)(16,89)(17,68)(18,65)(19,66)(20,67)(21,102)(22,103)(23,104)(24,101)(25,50)(26,51)(27,52)(28,49)(29,87)(30,88)(31,85)(32,86)(33,47)(34,48)(35,45)(36,46)(37,118)(38,119)(39,120)(40,117)(53,130)(54,131)(55,132)(56,129)(57,125)(58,126)(59,127)(60,128)(69,107)(70,108)(71,105)(72,106)(73,134)(74,135)(75,136)(76,133)(77,138)(78,139)(79,140)(80,137)(81,143)(82,144)(83,141)(84,142)(93,124)(94,121)(95,122)(96,123)(97,113)(98,114)(99,115)(100,116), (1,40,42)(2,43,37)(3,38,44)(4,41,39)(5,106,115)(6,116,107)(7,108,113)(8,114,105)(9,112,117)(10,118,109)(11,110,119)(12,120,111)(13,96,65)(14,66,93)(15,94,67)(16,68,95)(17,122,89)(18,90,123)(19,124,91)(20,92,121)(21,125,129)(22,130,126)(23,127,131)(24,132,128)(25,32,140)(26,137,29)(27,30,138)(28,139,31)(33,73,141)(34,142,74)(35,75,143)(36,144,76)(45,136,81)(46,82,133)(47,134,83)(48,84,135)(49,78,85)(50,86,79)(51,80,87)(52,88,77)(53,58,103)(54,104,59)(55,60,101)(56,102,57)(61,72,99)(62,100,69)(63,70,97)(64,98,71), (1,134,29)(2,30,135)(3,136,31)(4,32,133)(5,104,95)(6,96,101)(7,102,93)(8,94,103)(9,33,80)(10,77,34)(11,35,78)(12,79,36)(13,60,107)(14,108,57)(15,58,105)(16,106,59)(17,99,131)(18,132,100)(19,97,129)(20,130,98)(21,124,63)(22,64,121)(23,122,61)(24,62,123)(25,82,39)(26,40,83)(27,84,37)(28,38,81)(41,140,46)(42,47,137)(43,138,48)(44,45,139)(49,119,143)(50,144,120)(51,117,141)(52,142,118)(53,114,67)(54,68,115)(55,116,65)(56,66,113)(69,90,128)(70,125,91)(71,92,126)(72,127,89)(73,87,112)(74,109,88)(75,85,110)(76,111,86), (1,26,47)(2,48,27)(3,28,45)(4,46,25)(5,54,16)(6,13,55)(7,56,14)(8,15,53)(9,87,141)(10,142,88)(11,85,143)(12,144,86)(17,72,23)(18,24,69)(19,70,21)(20,22,71)(29,83,42)(30,43,84)(31,81,44)(32,41,82)(33,112,51)(34,52,109)(35,110,49)(36,50,111)(37,135,138)(38,139,136)(39,133,140)(40,137,134)(57,93,113)(58,114,94)(59,95,115)(60,116,96)(61,131,89)(62,90,132)(63,129,91)(64,92,130)(65,101,107)(66,108,102)(67,103,105)(68,106,104)(73,117,80)(74,77,118)(75,119,78)(76,79,120)(97,125,124)(98,121,126)(99,127,122)(100,123,128), (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), (1,57)(2,60)(3,59)(4,58)(5,139)(6,138)(7,137)(8,140)(9,21)(10,24)(11,23)(12,22)(13,135)(14,134)(15,133)(16,136)(17,143)(18,142)(19,141)(20,144)(25,114)(26,113)(27,116)(28,115)(29,108)(30,107)(31,106)(32,105)(33,124)(34,123)(35,122)(36,121)(37,55)(38,54)(39,53)(40,56)(41,103)(42,102)(43,101)(44,104)(45,95)(46,94)(47,93)(48,96)(49,99)(50,98)(51,97)(52,100)(61,78)(62,77)(63,80)(64,79)(65,84)(66,83)(67,82)(68,81)(69,88)(70,87)(71,86)(72,85)(73,91)(74,90)(75,89)(76,92)(109,128)(110,127)(111,126)(112,125)(117,129)(118,132)(119,131)(120,130) );
G=PermutationGroup([[(1,112),(2,109),(3,110),(4,111),(5,61),(6,62),(7,63),(8,64),(9,42),(10,43),(11,44),(12,41),(13,90),(14,91),(15,92),(16,89),(17,68),(18,65),(19,66),(20,67),(21,102),(22,103),(23,104),(24,101),(25,50),(26,51),(27,52),(28,49),(29,87),(30,88),(31,85),(32,86),(33,47),(34,48),(35,45),(36,46),(37,118),(38,119),(39,120),(40,117),(53,130),(54,131),(55,132),(56,129),(57,125),(58,126),(59,127),(60,128),(69,107),(70,108),(71,105),(72,106),(73,134),(74,135),(75,136),(76,133),(77,138),(78,139),(79,140),(80,137),(81,143),(82,144),(83,141),(84,142),(93,124),(94,121),(95,122),(96,123),(97,113),(98,114),(99,115),(100,116)], [(1,40,42),(2,43,37),(3,38,44),(4,41,39),(5,106,115),(6,116,107),(7,108,113),(8,114,105),(9,112,117),(10,118,109),(11,110,119),(12,120,111),(13,96,65),(14,66,93),(15,94,67),(16,68,95),(17,122,89),(18,90,123),(19,124,91),(20,92,121),(21,125,129),(22,130,126),(23,127,131),(24,132,128),(25,32,140),(26,137,29),(27,30,138),(28,139,31),(33,73,141),(34,142,74),(35,75,143),(36,144,76),(45,136,81),(46,82,133),(47,134,83),(48,84,135),(49,78,85),(50,86,79),(51,80,87),(52,88,77),(53,58,103),(54,104,59),(55,60,101),(56,102,57),(61,72,99),(62,100,69),(63,70,97),(64,98,71)], [(1,134,29),(2,30,135),(3,136,31),(4,32,133),(5,104,95),(6,96,101),(7,102,93),(8,94,103),(9,33,80),(10,77,34),(11,35,78),(12,79,36),(13,60,107),(14,108,57),(15,58,105),(16,106,59),(17,99,131),(18,132,100),(19,97,129),(20,130,98),(21,124,63),(22,64,121),(23,122,61),(24,62,123),(25,82,39),(26,40,83),(27,84,37),(28,38,81),(41,140,46),(42,47,137),(43,138,48),(44,45,139),(49,119,143),(50,144,120),(51,117,141),(52,142,118),(53,114,67),(54,68,115),(55,116,65),(56,66,113),(69,90,128),(70,125,91),(71,92,126),(72,127,89),(73,87,112),(74,109,88),(75,85,110),(76,111,86)], [(1,26,47),(2,48,27),(3,28,45),(4,46,25),(5,54,16),(6,13,55),(7,56,14),(8,15,53),(9,87,141),(10,142,88),(11,85,143),(12,144,86),(17,72,23),(18,24,69),(19,70,21),(20,22,71),(29,83,42),(30,43,84),(31,81,44),(32,41,82),(33,112,51),(34,52,109),(35,110,49),(36,50,111),(37,135,138),(38,139,136),(39,133,140),(40,137,134),(57,93,113),(58,114,94),(59,95,115),(60,116,96),(61,131,89),(62,90,132),(63,129,91),(64,92,130),(65,101,107),(66,108,102),(67,103,105),(68,106,104),(73,117,80),(74,77,118),(75,119,78),(76,79,120),(97,125,124),(98,121,126),(99,127,122),(100,123,128)], [(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)], [(1,57),(2,60),(3,59),(4,58),(5,139),(6,138),(7,137),(8,140),(9,21),(10,24),(11,23),(12,22),(13,135),(14,134),(15,133),(16,136),(17,143),(18,142),(19,141),(20,144),(25,114),(26,113),(27,116),(28,115),(29,108),(30,107),(31,106),(32,105),(33,124),(34,123),(35,122),(36,121),(37,55),(38,54),(39,53),(40,56),(41,103),(42,102),(43,101),(44,104),(45,95),(46,94),(47,93),(48,96),(49,99),(50,98),(51,97),(52,100),(61,78),(62,77),(63,80),(64,79),(65,84),(66,83),(67,82),(68,81),(69,88),(70,87),(71,86),(72,85),(73,91),(74,90),(75,89),(76,92),(109,128),(110,127),(111,126),(112,125),(117,129),(118,132),(119,131),(120,130)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 6A | ··· | 6O | 6P | ··· | 6AA | 6AB | ··· | 6AQ | 6AR | 6AS | 6AT | 6AU |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 54 | 54 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 18 | 18 | 18 | 18 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | C3⋊D4 | S32 | D6⋊S3 | C2×S32 |
| kernel | C2×C33⋊6D4 | C33⋊6D4 | C2×C33⋊5C4 | S3×C62 | C2×C6×C3⋊S3 | S3×C2×C6 | C22×C3⋊S3 | C32×C6 | S3×C6 | C2×C3⋊S3 | C62 | C3×C6 | C2×C6 | C6 | C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 1 | 2 | 8 | 2 | 5 | 20 | 4 | 8 | 4 |
Matrix representation of C2×C33⋊6D4 ►in GL6(𝔽13)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,4,0,0,0,0,2,11,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,9,0,0,0,0,4,2,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C2×C33⋊6D4 in GAP, Magma, Sage, TeX
C_2\times C_3^3\rtimes_6D_4
% in TeX
G:=Group("C2xC3^3:6D4"); // GroupNames label
G:=SmallGroup(432,680);
// by ID
G=gap.SmallGroup(432,680);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=f^2=1,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,e*b*e^-1=b^-1,b*f=f*b,c*d=d*c,e*c*e^-1=c^-1,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations