direct product, metabelian, supersoluble, monomial
Aliases: S3×C32⋊4Q8, C12.56S32, C33⋊9(C2×Q8), C3⋊4(S3×Dic6), (S3×C12).9S3, (C3×S3)⋊2Dic6, (S3×C6).39D6, (S3×C32)⋊4Q8, C32⋊14(S3×Q8), C33⋊4Q8⋊5C2, C33⋊8Q8⋊5C2, (C3×C12).138D6, C3⋊Dic3.34D6, C32⋊8(C2×Dic6), (C3×Dic3).34D6, (C32×C6).38C23, C33⋊5C4.7C22, (C32×C12).40C22, (C32×Dic3).21C22, C6.48(C2×S32), C4.5(S3×C3⋊S3), (S3×C3×C12).4C2, D6.8(C2×C3⋊S3), C12.21(C2×C3⋊S3), C6.1(C22×C3⋊S3), (C4×S3).1(C3⋊S3), C3⋊1(C2×C32⋊4Q8), (S3×C3×C6).23C22, Dic3.7(C2×C3⋊S3), (S3×C3⋊Dic3).2C2, (C3×C32⋊4Q8)⋊8C2, (C3×C6).97(C22×S3), (C3×C3⋊Dic3).17C22, C2.5(C2×S3×C3⋊S3), SmallGroup(432,660)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C32⋊4Q8
G = < a,b,c,d,e,f | a3=b2=c3=d3=e4=1, f2=e2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=c-1, de=ed, fdf-1=d-1, fef-1=e-1 >
Subgroups: 1328 in 276 conjugacy classes, 80 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, C32, C32, C32, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, C2×Dic6, S3×Q8, S3×C32, C32×C6, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C33⋊5C4, C32×C12, S3×C3×C6, S3×Dic6, C2×C32⋊4Q8, S3×C3⋊Dic3, C33⋊4Q8, S3×C3×C12, C3×C32⋊4Q8, C33⋊8Q8, S3×C32⋊4Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C3⋊S3, Dic6, C22×S3, S32, C2×C3⋊S3, C2×Dic6, S3×Q8, C32⋊4Q8, C2×S32, C22×C3⋊S3, S3×C3⋊S3, S3×Dic6, C2×C32⋊4Q8, C2×S3×C3⋊S3, S3×C32⋊4Q8
►On 144 points
Generators in S144
(1 29 21)(2 30 22)(3 31 23)(4 32 24)(5 50 34)(6 51 35)(7 52 36)(8 49 33)(9 115 141)(10 116 142)(11 113 143)(12 114 144)(13 37 59)(14 38 60)(15 39 57)(16 40 58)(17 125 117)(18 126 118)(19 127 119)(20 128 120)(25 140 133)(26 137 134)(27 138 135)(28 139 136)(41 78 71)(42 79 72)(43 80 69)(44 77 70)(45 53 75)(46 54 76)(47 55 73)(48 56 74)(61 68 96)(62 65 93)(63 66 94)(64 67 95)(81 89 110)(82 90 111)(83 91 112)(84 92 109)(85 108 124)(86 105 121)(87 106 122)(88 107 123)(97 130 104)(98 131 101)(99 132 102)(100 129 103)
(1 110)(2 111)(3 112)(4 109)(5 85)(6 86)(7 87)(8 88)(9 78)(10 79)(11 80)(12 77)(13 100)(14 97)(15 98)(16 99)(17 73)(18 74)(19 75)(20 76)(21 81)(22 82)(23 83)(24 84)(25 94)(26 95)(27 96)(28 93)(29 89)(30 90)(31 91)(32 92)(33 107)(34 108)(35 105)(36 106)(37 103)(38 104)(39 101)(40 102)(41 115)(42 116)(43 113)(44 114)(45 119)(46 120)(47 117)(48 118)(49 123)(50 124)(51 121)(52 122)(53 127)(54 128)(55 125)(56 126)(57 131)(58 132)(59 129)(60 130)(61 135)(62 136)(63 133)(64 134)(65 139)(66 140)(67 137)(68 138)(69 143)(70 144)(71 141)(72 142)
(1 40 33)(2 37 34)(3 38 35)(4 39 36)(5 30 59)(6 31 60)(7 32 57)(8 29 58)(9 135 126)(10 136 127)(11 133 128)(12 134 125)(13 50 22)(14 51 23)(15 52 24)(16 49 21)(17 144 137)(18 141 138)(19 142 139)(20 143 140)(25 120 113)(26 117 114)(27 118 115)(28 119 116)(41 96 48)(42 93 45)(43 94 46)(44 95 47)(53 79 62)(54 80 63)(55 77 64)(56 78 61)(65 75 72)(66 76 69)(67 73 70)(68 74 71)(81 99 123)(82 100 124)(83 97 121)(84 98 122)(85 90 129)(86 91 130)(87 92 131)(88 89 132)(101 106 109)(102 107 110)(103 108 111)(104 105 112)
(1 16 8)(2 13 5)(3 14 6)(4 15 7)(9 138 118)(10 139 119)(11 140 120)(12 137 117)(17 114 134)(18 115 135)(19 116 136)(20 113 133)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 128 143)(26 125 144)(27 126 141)(28 127 142)(29 40 49)(30 37 50)(31 38 51)(32 39 52)(41 61 74)(42 62 75)(43 63 76)(44 64 73)(45 79 65)(46 80 66)(47 77 67)(48 78 68)(53 72 93)(54 69 94)(55 70 95)(56 71 96)(81 132 107)(82 129 108)(83 130 105)(84 131 106)(85 111 100)(86 112 97)(87 109 98)(88 110 99)(89 102 123)(90 103 124)(91 104 121)(92 101 122)
(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 76 3 74)(2 75 4 73)(5 42 7 44)(6 41 8 43)(9 123 11 121)(10 122 12 124)(13 62 15 64)(14 61 16 63)(17 111 19 109)(18 110 20 112)(21 54 23 56)(22 53 24 55)(25 130 27 132)(26 129 28 131)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 65 39 67)(38 68 40 66)(49 80 51 78)(50 79 52 77)(57 95 59 93)(58 94 60 96)(81 128 83 126)(82 127 84 125)(85 116 87 114)(86 115 88 113)(89 120 91 118)(90 119 92 117)(97 135 99 133)(98 134 100 136)(101 137 103 139)(102 140 104 138)(105 141 107 143)(106 144 108 142)
G:=sub<Sym(144)| (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,50,34)(6,51,35)(7,52,36)(8,49,33)(9,115,141)(10,116,142)(11,113,143)(12,114,144)(13,37,59)(14,38,60)(15,39,57)(16,40,58)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(25,140,133)(26,137,134)(27,138,135)(28,139,136)(41,78,71)(42,79,72)(43,80,69)(44,77,70)(45,53,75)(46,54,76)(47,55,73)(48,56,74)(61,68,96)(62,65,93)(63,66,94)(64,67,95)(81,89,110)(82,90,111)(83,91,112)(84,92,109)(85,108,124)(86,105,121)(87,106,122)(88,107,123)(97,130,104)(98,131,101)(99,132,102)(100,129,103), (1,110)(2,111)(3,112)(4,109)(5,85)(6,86)(7,87)(8,88)(9,78)(10,79)(11,80)(12,77)(13,100)(14,97)(15,98)(16,99)(17,73)(18,74)(19,75)(20,76)(21,81)(22,82)(23,83)(24,84)(25,94)(26,95)(27,96)(28,93)(29,89)(30,90)(31,91)(32,92)(33,107)(34,108)(35,105)(36,106)(37,103)(38,104)(39,101)(40,102)(41,115)(42,116)(43,113)(44,114)(45,119)(46,120)(47,117)(48,118)(49,123)(50,124)(51,121)(52,122)(53,127)(54,128)(55,125)(56,126)(57,131)(58,132)(59,129)(60,130)(61,135)(62,136)(63,133)(64,134)(65,139)(66,140)(67,137)(68,138)(69,143)(70,144)(71,141)(72,142), (1,40,33)(2,37,34)(3,38,35)(4,39,36)(5,30,59)(6,31,60)(7,32,57)(8,29,58)(9,135,126)(10,136,127)(11,133,128)(12,134,125)(13,50,22)(14,51,23)(15,52,24)(16,49,21)(17,144,137)(18,141,138)(19,142,139)(20,143,140)(25,120,113)(26,117,114)(27,118,115)(28,119,116)(41,96,48)(42,93,45)(43,94,46)(44,95,47)(53,79,62)(54,80,63)(55,77,64)(56,78,61)(65,75,72)(66,76,69)(67,73,70)(68,74,71)(81,99,123)(82,100,124)(83,97,121)(84,98,122)(85,90,129)(86,91,130)(87,92,131)(88,89,132)(101,106,109)(102,107,110)(103,108,111)(104,105,112), (1,16,8)(2,13,5)(3,14,6)(4,15,7)(9,138,118)(10,139,119)(11,140,120)(12,137,117)(17,114,134)(18,115,135)(19,116,136)(20,113,133)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,128,143)(26,125,144)(27,126,141)(28,127,142)(29,40,49)(30,37,50)(31,38,51)(32,39,52)(41,61,74)(42,62,75)(43,63,76)(44,64,73)(45,79,65)(46,80,66)(47,77,67)(48,78,68)(53,72,93)(54,69,94)(55,70,95)(56,71,96)(81,132,107)(82,129,108)(83,130,105)(84,131,106)(85,111,100)(86,112,97)(87,109,98)(88,110,99)(89,102,123)(90,103,124)(91,104,121)(92,101,122), (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), (1,76,3,74)(2,75,4,73)(5,42,7,44)(6,41,8,43)(9,123,11,121)(10,122,12,124)(13,62,15,64)(14,61,16,63)(17,111,19,109)(18,110,20,112)(21,54,23,56)(22,53,24,55)(25,130,27,132)(26,129,28,131)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,65,39,67)(38,68,40,66)(49,80,51,78)(50,79,52,77)(57,95,59,93)(58,94,60,96)(81,128,83,126)(82,127,84,125)(85,116,87,114)(86,115,88,113)(89,120,91,118)(90,119,92,117)(97,135,99,133)(98,134,100,136)(101,137,103,139)(102,140,104,138)(105,141,107,143)(106,144,108,142)>;
G:=Group( (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,50,34)(6,51,35)(7,52,36)(8,49,33)(9,115,141)(10,116,142)(11,113,143)(12,114,144)(13,37,59)(14,38,60)(15,39,57)(16,40,58)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(25,140,133)(26,137,134)(27,138,135)(28,139,136)(41,78,71)(42,79,72)(43,80,69)(44,77,70)(45,53,75)(46,54,76)(47,55,73)(48,56,74)(61,68,96)(62,65,93)(63,66,94)(64,67,95)(81,89,110)(82,90,111)(83,91,112)(84,92,109)(85,108,124)(86,105,121)(87,106,122)(88,107,123)(97,130,104)(98,131,101)(99,132,102)(100,129,103), (1,110)(2,111)(3,112)(4,109)(5,85)(6,86)(7,87)(8,88)(9,78)(10,79)(11,80)(12,77)(13,100)(14,97)(15,98)(16,99)(17,73)(18,74)(19,75)(20,76)(21,81)(22,82)(23,83)(24,84)(25,94)(26,95)(27,96)(28,93)(29,89)(30,90)(31,91)(32,92)(33,107)(34,108)(35,105)(36,106)(37,103)(38,104)(39,101)(40,102)(41,115)(42,116)(43,113)(44,114)(45,119)(46,120)(47,117)(48,118)(49,123)(50,124)(51,121)(52,122)(53,127)(54,128)(55,125)(56,126)(57,131)(58,132)(59,129)(60,130)(61,135)(62,136)(63,133)(64,134)(65,139)(66,140)(67,137)(68,138)(69,143)(70,144)(71,141)(72,142), (1,40,33)(2,37,34)(3,38,35)(4,39,36)(5,30,59)(6,31,60)(7,32,57)(8,29,58)(9,135,126)(10,136,127)(11,133,128)(12,134,125)(13,50,22)(14,51,23)(15,52,24)(16,49,21)(17,144,137)(18,141,138)(19,142,139)(20,143,140)(25,120,113)(26,117,114)(27,118,115)(28,119,116)(41,96,48)(42,93,45)(43,94,46)(44,95,47)(53,79,62)(54,80,63)(55,77,64)(56,78,61)(65,75,72)(66,76,69)(67,73,70)(68,74,71)(81,99,123)(82,100,124)(83,97,121)(84,98,122)(85,90,129)(86,91,130)(87,92,131)(88,89,132)(101,106,109)(102,107,110)(103,108,111)(104,105,112), (1,16,8)(2,13,5)(3,14,6)(4,15,7)(9,138,118)(10,139,119)(11,140,120)(12,137,117)(17,114,134)(18,115,135)(19,116,136)(20,113,133)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,128,143)(26,125,144)(27,126,141)(28,127,142)(29,40,49)(30,37,50)(31,38,51)(32,39,52)(41,61,74)(42,62,75)(43,63,76)(44,64,73)(45,79,65)(46,80,66)(47,77,67)(48,78,68)(53,72,93)(54,69,94)(55,70,95)(56,71,96)(81,132,107)(82,129,108)(83,130,105)(84,131,106)(85,111,100)(86,112,97)(87,109,98)(88,110,99)(89,102,123)(90,103,124)(91,104,121)(92,101,122), (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), (1,76,3,74)(2,75,4,73)(5,42,7,44)(6,41,8,43)(9,123,11,121)(10,122,12,124)(13,62,15,64)(14,61,16,63)(17,111,19,109)(18,110,20,112)(21,54,23,56)(22,53,24,55)(25,130,27,132)(26,129,28,131)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,65,39,67)(38,68,40,66)(49,80,51,78)(50,79,52,77)(57,95,59,93)(58,94,60,96)(81,128,83,126)(82,127,84,125)(85,116,87,114)(86,115,88,113)(89,120,91,118)(90,119,92,117)(97,135,99,133)(98,134,100,136)(101,137,103,139)(102,140,104,138)(105,141,107,143)(106,144,108,142) );
G=PermutationGroup([[(1,29,21),(2,30,22),(3,31,23),(4,32,24),(5,50,34),(6,51,35),(7,52,36),(8,49,33),(9,115,141),(10,116,142),(11,113,143),(12,114,144),(13,37,59),(14,38,60),(15,39,57),(16,40,58),(17,125,117),(18,126,118),(19,127,119),(20,128,120),(25,140,133),(26,137,134),(27,138,135),(28,139,136),(41,78,71),(42,79,72),(43,80,69),(44,77,70),(45,53,75),(46,54,76),(47,55,73),(48,56,74),(61,68,96),(62,65,93),(63,66,94),(64,67,95),(81,89,110),(82,90,111),(83,91,112),(84,92,109),(85,108,124),(86,105,121),(87,106,122),(88,107,123),(97,130,104),(98,131,101),(99,132,102),(100,129,103)], [(1,110),(2,111),(3,112),(4,109),(5,85),(6,86),(7,87),(8,88),(9,78),(10,79),(11,80),(12,77),(13,100),(14,97),(15,98),(16,99),(17,73),(18,74),(19,75),(20,76),(21,81),(22,82),(23,83),(24,84),(25,94),(26,95),(27,96),(28,93),(29,89),(30,90),(31,91),(32,92),(33,107),(34,108),(35,105),(36,106),(37,103),(38,104),(39,101),(40,102),(41,115),(42,116),(43,113),(44,114),(45,119),(46,120),(47,117),(48,118),(49,123),(50,124),(51,121),(52,122),(53,127),(54,128),(55,125),(56,126),(57,131),(58,132),(59,129),(60,130),(61,135),(62,136),(63,133),(64,134),(65,139),(66,140),(67,137),(68,138),(69,143),(70,144),(71,141),(72,142)], [(1,40,33),(2,37,34),(3,38,35),(4,39,36),(5,30,59),(6,31,60),(7,32,57),(8,29,58),(9,135,126),(10,136,127),(11,133,128),(12,134,125),(13,50,22),(14,51,23),(15,52,24),(16,49,21),(17,144,137),(18,141,138),(19,142,139),(20,143,140),(25,120,113),(26,117,114),(27,118,115),(28,119,116),(41,96,48),(42,93,45),(43,94,46),(44,95,47),(53,79,62),(54,80,63),(55,77,64),(56,78,61),(65,75,72),(66,76,69),(67,73,70),(68,74,71),(81,99,123),(82,100,124),(83,97,121),(84,98,122),(85,90,129),(86,91,130),(87,92,131),(88,89,132),(101,106,109),(102,107,110),(103,108,111),(104,105,112)], [(1,16,8),(2,13,5),(3,14,6),(4,15,7),(9,138,118),(10,139,119),(11,140,120),(12,137,117),(17,114,134),(18,115,135),(19,116,136),(20,113,133),(21,58,33),(22,59,34),(23,60,35),(24,57,36),(25,128,143),(26,125,144),(27,126,141),(28,127,142),(29,40,49),(30,37,50),(31,38,51),(32,39,52),(41,61,74),(42,62,75),(43,63,76),(44,64,73),(45,79,65),(46,80,66),(47,77,67),(48,78,68),(53,72,93),(54,69,94),(55,70,95),(56,71,96),(81,132,107),(82,129,108),(83,130,105),(84,131,106),(85,111,100),(86,112,97),(87,109,98),(88,110,99),(89,102,123),(90,103,124),(91,104,121),(92,101,122)], [(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)], [(1,76,3,74),(2,75,4,73),(5,42,7,44),(6,41,8,43),(9,123,11,121),(10,122,12,124),(13,62,15,64),(14,61,16,63),(17,111,19,109),(18,110,20,112),(21,54,23,56),(22,53,24,55),(25,130,27,132),(26,129,28,131),(29,46,31,48),(30,45,32,47),(33,69,35,71),(34,72,36,70),(37,65,39,67),(38,68,40,66),(49,80,51,78),(50,79,52,77),(57,95,59,93),(58,94,60,96),(81,128,83,126),(82,127,84,125),(85,116,87,114),(86,115,88,113),(89,120,91,118),(90,119,92,117),(97,135,99,133),(98,134,100,136),(101,137,103,139),(102,140,104,138),(105,141,107,143),(106,144,108,142)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3E | 3F | 3G | 3H | 3I | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6E | 6F | 6G | 6H | 6I | 6J | ··· | 6Q | 12A | ··· | 12H | 12I | ··· | 12Q | 12R | ··· | 12Y | 12Z | 12AA |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 6 | 18 | 18 | 54 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 36 | 36 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | Q8 | D6 | D6 | D6 | D6 | Dic6 | S32 | S3×Q8 | C2×S32 | S3×Dic6 |
| kernel | S3×C32⋊4Q8 | S3×C3⋊Dic3 | C33⋊4Q8 | S3×C3×C12 | C3×C32⋊4Q8 | C33⋊8Q8 | S3×C12 | C32⋊4Q8 | S3×C32 | C3×Dic3 | C3⋊Dic3 | C3×C12 | S3×C6 | C3×S3 | C12 | C32 | C6 | C3 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 2 | 4 | 2 | 5 | 4 | 16 | 4 | 1 | 4 | 8 |
Matrix representation of S3×C32⋊4Q8 ►in GL8(𝔽13)
| 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 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 | 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 |
| 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 | 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 |
| 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 | 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 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 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))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,3,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,4,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
S3×C32⋊4Q8 in GAP, Magma, Sage, TeX
S_3\times C_3^2\rtimes_4Q_8
% in TeX
G:=Group("S3xC3^2:4Q8"); // GroupNames label
G:=SmallGroup(432,660);
// by ID
G=gap.SmallGroup(432,660);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,135,58,571,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^3=e^4=1,f^2=e^2,b*a*b=a^-1,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,f*c*f^-1=c^-1,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations