direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×Dic3×C3⋊S3, C62.92D6, C6⋊1(S3×Dic3), (C6×Dic3)⋊7S3, (C3×Dic3)⋊15D6, C33⋊13(C22×C4), C33⋊5C4⋊10C22, (C3×C62).26C22, (C32×C6).55C23, C32⋊9(C22×Dic3), (C32×Dic3)⋊20C22, C6⋊3(C4×C3⋊S3), (C6×C3⋊S3)⋊8C4, C6.65(C2×S32), (C3×C6)⋊6(C4×S3), (C2×C6).39S32, C3⋊2(C2×S3×Dic3), C32⋊12(S3×C2×C4), (C32×C6)⋊7(C2×C4), (C2×C3⋊S3).54D6, (Dic3×C3×C6)⋊13C2, (C3×C6)⋊8(C2×Dic3), (C2×C33⋊5C4)⋊7C2, (C22×C3⋊S3).9S3, C6.18(C22×C3⋊S3), C22.12(S3×C3⋊S3), (C6×C3⋊S3).56C22, (C3×C6).110(C22×S3), C3⋊4(C2×C4×C3⋊S3), C2.3(C2×S3×C3⋊S3), (C2×C6×C3⋊S3).8C2, (C3×C3⋊S3)⋊10(C2×C4), (C2×C6).21(C2×C3⋊S3), SmallGroup(432,677)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C2×Dic3×C3⋊S3 |
Generators and relations for C2×Dic3×C3⋊S3
G = < a,b,c,d,e,f | a2=b6=d3=e3=f2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >
Subgroups: 1688 in 388 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22×C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C62, S3×C2×C4, C22×Dic3, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C33⋊5C4, C6×C3⋊S3, C3×C62, C2×S3×Dic3, C2×C4×C3⋊S3, Dic3×C3⋊S3, Dic3×C3×C6, C2×C33⋊5C4, C2×C6×C3⋊S3, C2×Dic3×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C3⋊S3, C4×S3, C2×Dic3, C22×S3, S32, C2×C3⋊S3, S3×C2×C4, C22×Dic3, S3×Dic3, C4×C3⋊S3, C2×S32, C22×C3⋊S3, S3×C3⋊S3, C2×S3×Dic3, C2×C4×C3⋊S3, Dic3×C3⋊S3, C2×S3×C3⋊S3, C2×Dic3×C3⋊S3
►On 144 points
Generators in S144
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 128)(8 129)(9 130)(10 131)(11 132)(12 127)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 78)(56 73)(57 74)(58 75)(59 76)(60 77)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 113)(96 114)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 139)(122 140)(123 141)(124 142)(125 143)(126 144)
(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 88 4 85)(2 87 5 90)(3 86 6 89)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 80 16 83)(14 79 17 82)(15 84 18 81)(19 95 22 92)(20 94 23 91)(21 93 24 96)(25 101 28 98)(26 100 29 97)(27 99 30 102)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 113 40 110)(38 112 41 109)(39 111 42 114)(43 119 46 116)(44 118 47 115)(45 117 48 120)(49 122 52 125)(50 121 53 124)(51 126 54 123)(55 131 58 128)(56 130 59 127)(57 129 60 132)(61 137 64 134)(62 136 65 133)(63 135 66 138)(67 140 70 143)(68 139 71 142)(69 144 72 141)
(1 19 13)(2 20 14)(3 21 15)(4 22 16)(5 23 17)(6 24 18)(7 139 137)(8 140 138)(9 141 133)(10 142 134)(11 143 135)(12 144 136)(25 32 40)(26 33 41)(27 34 42)(28 35 37)(29 36 38)(30 31 39)(43 58 50)(44 59 51)(45 60 52)(46 55 53)(47 56 54)(48 57 49)(61 75 68)(62 76 69)(63 77 70)(64 78 71)(65 73 72)(66 74 67)(79 87 94)(80 88 95)(81 89 96)(82 90 91)(83 85 92)(84 86 93)(97 105 112)(98 106 113)(99 107 114)(100 108 109)(101 103 110)(102 104 111)(115 130 123)(116 131 124)(117 132 125)(118 127 126)(119 128 121)(120 129 122)
(1 17 21)(2 18 22)(3 13 23)(4 14 24)(5 15 19)(6 16 20)(7 135 141)(8 136 142)(9 137 143)(10 138 144)(11 133 139)(12 134 140)(25 38 34)(26 39 35)(27 40 36)(28 41 31)(29 42 32)(30 37 33)(43 52 56)(44 53 57)(45 54 58)(46 49 59)(47 50 60)(48 51 55)(61 70 73)(62 71 74)(63 72 75)(64 67 76)(65 68 77)(66 69 78)(79 96 85)(80 91 86)(81 92 87)(82 93 88)(83 94 89)(84 95 90)(97 114 103)(98 109 104)(99 110 105)(100 111 106)(101 112 107)(102 113 108)(115 121 132)(116 122 127)(117 123 128)(118 124 129)(119 125 130)(120 126 131)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 113)(8 114)(9 109)(10 110)(11 111)(12 112)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 58)(20 59)(21 60)(22 55)(23 56)(24 57)(25 64)(26 65)(27 66)(28 61)(29 62)(30 63)(31 70)(32 71)(33 72)(34 67)(35 68)(36 69)(37 75)(38 76)(39 77)(40 78)(41 73)(42 74)(79 118)(80 119)(81 120)(82 115)(83 116)(84 117)(85 124)(86 125)(87 126)(88 121)(89 122)(90 123)(91 130)(92 131)(93 132)(94 127)(95 128)(96 129)(97 136)(98 137)(99 138)(100 133)(101 134)(102 135)(103 142)(104 143)(105 144)(106 139)(107 140)(108 141)
G:=sub<Sym(144)| (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,128)(8,129)(9,130)(10,131)(11,132)(12,127)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,78)(56,73)(57,74)(58,75)(59,76)(60,77)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (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), (1,88,4,85)(2,87,5,90)(3,86,6,89)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,80,16,83)(14,79,17,82)(15,84,18,81)(19,95,22,92)(20,94,23,91)(21,93,24,96)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,113,40,110)(38,112,41,109)(39,111,42,114)(43,119,46,116)(44,118,47,115)(45,117,48,120)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,131,58,128)(56,130,59,127)(57,129,60,132)(61,137,64,134)(62,136,65,133)(63,135,66,138)(67,140,70,143)(68,139,71,142)(69,144,72,141), (1,19,13)(2,20,14)(3,21,15)(4,22,16)(5,23,17)(6,24,18)(7,139,137)(8,140,138)(9,141,133)(10,142,134)(11,143,135)(12,144,136)(25,32,40)(26,33,41)(27,34,42)(28,35,37)(29,36,38)(30,31,39)(43,58,50)(44,59,51)(45,60,52)(46,55,53)(47,56,54)(48,57,49)(61,75,68)(62,76,69)(63,77,70)(64,78,71)(65,73,72)(66,74,67)(79,87,94)(80,88,95)(81,89,96)(82,90,91)(83,85,92)(84,86,93)(97,105,112)(98,106,113)(99,107,114)(100,108,109)(101,103,110)(102,104,111)(115,130,123)(116,131,124)(117,132,125)(118,127,126)(119,128,121)(120,129,122), (1,17,21)(2,18,22)(3,13,23)(4,14,24)(5,15,19)(6,16,20)(7,135,141)(8,136,142)(9,137,143)(10,138,144)(11,133,139)(12,134,140)(25,38,34)(26,39,35)(27,40,36)(28,41,31)(29,42,32)(30,37,33)(43,52,56)(44,53,57)(45,54,58)(46,49,59)(47,50,60)(48,51,55)(61,70,73)(62,71,74)(63,72,75)(64,67,76)(65,68,77)(66,69,78)(79,96,85)(80,91,86)(81,92,87)(82,93,88)(83,94,89)(84,95,90)(97,114,103)(98,109,104)(99,110,105)(100,111,106)(101,112,107)(102,113,108)(115,121,132)(116,122,127)(117,123,128)(118,124,129)(119,125,130)(120,126,131), (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,113)(8,114)(9,109)(10,110)(11,111)(12,112)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,58)(20,59)(21,60)(22,55)(23,56)(24,57)(25,64)(26,65)(27,66)(28,61)(29,62)(30,63)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,75)(38,76)(39,77)(40,78)(41,73)(42,74)(79,118)(80,119)(81,120)(82,115)(83,116)(84,117)(85,124)(86,125)(87,126)(88,121)(89,122)(90,123)(91,130)(92,131)(93,132)(94,127)(95,128)(96,129)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,142)(104,143)(105,144)(106,139)(107,140)(108,141)>;
G:=Group( (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,128)(8,129)(9,130)(10,131)(11,132)(12,127)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,78)(56,73)(57,74)(58,75)(59,76)(60,77)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,113)(96,114)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144), (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), (1,88,4,85)(2,87,5,90)(3,86,6,89)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,80,16,83)(14,79,17,82)(15,84,18,81)(19,95,22,92)(20,94,23,91)(21,93,24,96)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,113,40,110)(38,112,41,109)(39,111,42,114)(43,119,46,116)(44,118,47,115)(45,117,48,120)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,131,58,128)(56,130,59,127)(57,129,60,132)(61,137,64,134)(62,136,65,133)(63,135,66,138)(67,140,70,143)(68,139,71,142)(69,144,72,141), (1,19,13)(2,20,14)(3,21,15)(4,22,16)(5,23,17)(6,24,18)(7,139,137)(8,140,138)(9,141,133)(10,142,134)(11,143,135)(12,144,136)(25,32,40)(26,33,41)(27,34,42)(28,35,37)(29,36,38)(30,31,39)(43,58,50)(44,59,51)(45,60,52)(46,55,53)(47,56,54)(48,57,49)(61,75,68)(62,76,69)(63,77,70)(64,78,71)(65,73,72)(66,74,67)(79,87,94)(80,88,95)(81,89,96)(82,90,91)(83,85,92)(84,86,93)(97,105,112)(98,106,113)(99,107,114)(100,108,109)(101,103,110)(102,104,111)(115,130,123)(116,131,124)(117,132,125)(118,127,126)(119,128,121)(120,129,122), (1,17,21)(2,18,22)(3,13,23)(4,14,24)(5,15,19)(6,16,20)(7,135,141)(8,136,142)(9,137,143)(10,138,144)(11,133,139)(12,134,140)(25,38,34)(26,39,35)(27,40,36)(28,41,31)(29,42,32)(30,37,33)(43,52,56)(44,53,57)(45,54,58)(46,49,59)(47,50,60)(48,51,55)(61,70,73)(62,71,74)(63,72,75)(64,67,76)(65,68,77)(66,69,78)(79,96,85)(80,91,86)(81,92,87)(82,93,88)(83,94,89)(84,95,90)(97,114,103)(98,109,104)(99,110,105)(100,111,106)(101,112,107)(102,113,108)(115,121,132)(116,122,127)(117,123,128)(118,124,129)(119,125,130)(120,126,131), (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,113)(8,114)(9,109)(10,110)(11,111)(12,112)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,58)(20,59)(21,60)(22,55)(23,56)(24,57)(25,64)(26,65)(27,66)(28,61)(29,62)(30,63)(31,70)(32,71)(33,72)(34,67)(35,68)(36,69)(37,75)(38,76)(39,77)(40,78)(41,73)(42,74)(79,118)(80,119)(81,120)(82,115)(83,116)(84,117)(85,124)(86,125)(87,126)(88,121)(89,122)(90,123)(91,130)(92,131)(93,132)(94,127)(95,128)(96,129)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,142)(104,143)(105,144)(106,139)(107,140)(108,141) );
G=PermutationGroup([[(1,35),(2,36),(3,31),(4,32),(5,33),(6,34),(7,128),(8,129),(9,130),(10,131),(11,132),(12,127),(13,28),(14,29),(15,30),(16,25),(17,26),(18,27),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,78),(56,73),(57,74),(58,75),(59,76),(60,77),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,113),(96,114),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(121,139),(122,140),(123,141),(124,142),(125,143),(126,144)], [(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)], [(1,88,4,85),(2,87,5,90),(3,86,6,89),(7,78,10,75),(8,77,11,74),(9,76,12,73),(13,80,16,83),(14,79,17,82),(15,84,18,81),(19,95,22,92),(20,94,23,91),(21,93,24,96),(25,101,28,98),(26,100,29,97),(27,99,30,102),(31,104,34,107),(32,103,35,106),(33,108,36,105),(37,113,40,110),(38,112,41,109),(39,111,42,114),(43,119,46,116),(44,118,47,115),(45,117,48,120),(49,122,52,125),(50,121,53,124),(51,126,54,123),(55,131,58,128),(56,130,59,127),(57,129,60,132),(61,137,64,134),(62,136,65,133),(63,135,66,138),(67,140,70,143),(68,139,71,142),(69,144,72,141)], [(1,19,13),(2,20,14),(3,21,15),(4,22,16),(5,23,17),(6,24,18),(7,139,137),(8,140,138),(9,141,133),(10,142,134),(11,143,135),(12,144,136),(25,32,40),(26,33,41),(27,34,42),(28,35,37),(29,36,38),(30,31,39),(43,58,50),(44,59,51),(45,60,52),(46,55,53),(47,56,54),(48,57,49),(61,75,68),(62,76,69),(63,77,70),(64,78,71),(65,73,72),(66,74,67),(79,87,94),(80,88,95),(81,89,96),(82,90,91),(83,85,92),(84,86,93),(97,105,112),(98,106,113),(99,107,114),(100,108,109),(101,103,110),(102,104,111),(115,130,123),(116,131,124),(117,132,125),(118,127,126),(119,128,121),(120,129,122)], [(1,17,21),(2,18,22),(3,13,23),(4,14,24),(5,15,19),(6,16,20),(7,135,141),(8,136,142),(9,137,143),(10,138,144),(11,133,139),(12,134,140),(25,38,34),(26,39,35),(27,40,36),(28,41,31),(29,42,32),(30,37,33),(43,52,56),(44,53,57),(45,54,58),(46,49,59),(47,50,60),(48,51,55),(61,70,73),(62,71,74),(63,72,75),(64,67,76),(65,68,77),(66,69,78),(79,96,85),(80,91,86),(81,92,87),(82,93,88),(83,94,89),(84,95,90),(97,114,103),(98,109,104),(99,110,105),(100,111,106),(101,112,107),(102,113,108),(115,121,132),(116,122,127),(117,123,128),(118,124,129),(119,125,130),(120,126,131)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,49),(7,113),(8,114),(9,109),(10,110),(11,111),(12,112),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,58),(20,59),(21,60),(22,55),(23,56),(24,57),(25,64),(26,65),(27,66),(28,61),(29,62),(30,63),(31,70),(32,71),(33,72),(34,67),(35,68),(36,69),(37,75),(38,76),(39,77),(40,78),(41,73),(42,74),(79,118),(80,119),(81,120),(82,115),(83,116),(84,117),(85,124),(86,125),(87,126),(88,121),(89,122),(90,123),(91,130),(92,131),(93,132),(94,127),(95,128),(96,129),(97,136),(98,137),(99,138),(100,133),(101,134),(102,135),(103,142),(104,143),(105,144),(106,139),(107,140),(108,141)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6O | 6P | ··· | 6AA | 6AB | 6AC | 6AD | 6AE | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D6 | Dic3 | D6 | D6 | C4×S3 | S32 | S3×Dic3 | C2×S32 |
| kernel | C2×Dic3×C3⋊S3 | Dic3×C3⋊S3 | Dic3×C3×C6 | C2×C33⋊5C4 | C2×C6×C3⋊S3 | C6×C3⋊S3 | C6×Dic3 | C22×C3⋊S3 | C3×Dic3 | C2×C3⋊S3 | C2×C3⋊S3 | C62 | C3×C6 | C2×C6 | C6 | C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 1 | 8 | 4 | 2 | 5 | 16 | 4 | 8 | 4 |
Matrix representation of C2×Dic3×C3⋊S3 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 10 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[10,7,0,0,0,0,0,0,10,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
C2×Dic3×C3⋊S3 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3\times C_3\rtimes S_3 % in TeX
G:=Group("C2xDic3xC3:S3"); // GroupNames label
G:=SmallGroup(432,677);
// by ID
G=gap.SmallGroup(432,677);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^6=d^3=e^3=f^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,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,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations