direct product, metabelian, supersoluble, monomial
Aliases: S3×Q8×C32, C12.6C62, D6.5C62, Dic3.4C62, (S3×C12).7C6, C33⋊19(C2×Q8), C12.62(S3×C6), Dic6⋊4(C3×C6), (C3×Dic6)⋊9C6, C2.8(S3×C62), C6.7(C2×C62), (Q8×C33)⋊4C2, C32⋊10(C6×Q8), (C3×C12).189D6, (Q8×C32)⋊14C6, (C32×Dic6)⋊15C2, (C32×C6).81C23, (C32×C12).52C22, (C32×Dic3).34C22, C3⋊2(Q8×C3×C6), C4.6(S3×C3×C6), C6.79(S3×C2×C6), (S3×C3×C12).5C2, (C3×Q8)⋊4(C3×C6), (C4×S3).1(C3×C6), (S3×C6).23(C2×C6), (C3×C12).56(C2×C6), (S3×C3×C6).34C22, (C3×C6).55(C22×C6), (C3×C6).200(C22×S3), (C3×Dic3).18(C2×C6), SmallGroup(432,706)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Q8×C32
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 480 in 276 conjugacy classes, 150 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, Q8, C32, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, S3×Q8, C6×Q8, S3×C32, C32×C6, C3×Dic6, S3×C12, C6×C12, Q8×C32, Q8×C32, Q8×C32, C32×Dic3, C32×C12, S3×C3×C6, C3×S3×Q8, Q8×C3×C6, C32×Dic6, S3×C3×C12, Q8×C33, S3×Q8×C32
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, C32, D6, C2×C6, C2×Q8, C3×S3, C3×C6, C3×Q8, C22×S3, C22×C6, S3×C6, C62, S3×Q8, C6×Q8, S3×C32, Q8×C32, S3×C2×C6, C2×C62, S3×C3×C6, C3×S3×Q8, Q8×C3×C6, S3×C62, S3×Q8×C32
►On 144 points
Generators in S144
(1 59 7)(2 60 8)(3 57 5)(4 58 6)(9 30 27)(10 31 28)(11 32 25)(12 29 26)(13 39 34)(14 40 35)(15 37 36)(16 38 33)(17 144 123)(18 141 124)(19 142 121)(20 143 122)(21 113 118)(22 114 119)(23 115 120)(24 116 117)(41 46 95)(42 47 96)(43 48 93)(44 45 94)(49 75 70)(50 76 71)(51 73 72)(52 74 69)(53 66 62)(54 67 63)(55 68 64)(56 65 61)(77 131 82)(78 132 83)(79 129 84)(80 130 81)(85 103 99)(86 104 100)(87 101 97)(88 102 98)(89 112 107)(90 109 108)(91 110 105)(92 111 106)(125 138 134)(126 139 135)(127 140 136)(128 137 133)
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 137)(18 23 138)(19 24 139)(20 21 140)(25 39 58)(26 40 59)(27 37 60)(28 38 57)(41 64 71)(42 61 72)(43 62 69)(44 63 70)(45 54 49)(46 55 50)(47 56 51)(48 53 52)(65 73 96)(66 74 93)(67 75 94)(68 76 95)(77 88 89)(78 85 90)(79 86 91)(80 87 92)(81 97 106)(82 98 107)(83 99 108)(84 100 105)(101 111 130)(102 112 131)(103 109 132)(104 110 129)(113 136 143)(114 133 144)(115 134 141)(116 135 142)(117 126 121)(118 127 122)(119 128 123)(120 125 124)
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 137 22)(18 138 23)(19 139 24)(20 140 21)(25 39 58)(26 40 59)(27 37 60)(28 38 57)(41 64 71)(42 61 72)(43 62 69)(44 63 70)(45 54 49)(46 55 50)(47 56 51)(48 53 52)(65 73 96)(66 74 93)(67 75 94)(68 76 95)(77 89 88)(78 90 85)(79 91 86)(80 92 87)(81 106 97)(82 107 98)(83 108 99)(84 105 100)(101 130 111)(102 131 112)(103 132 109)(104 129 110)(113 143 136)(114 144 133)(115 141 134)(116 142 135)(117 121 126)(118 122 127)(119 123 128)(120 124 125)
(1 77)(2 78)(3 79)(4 80)(5 84)(6 81)(7 82)(8 83)(9 99)(10 100)(11 97)(12 98)(13 92)(14 89)(15 90)(16 91)(17 73)(18 74)(19 75)(20 76)(21 95)(22 96)(23 93)(24 94)(25 101)(26 102)(27 103)(28 104)(29 88)(30 85)(31 86)(32 87)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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 46 3 48)(2 45 4 47)(5 43 7 41)(6 42 8 44)(9 63 11 61)(10 62 12 64)(13 51 15 49)(14 50 16 52)(17 109 19 111)(18 112 20 110)(21 129 23 131)(22 132 24 130)(25 65 27 67)(26 68 28 66)(29 55 31 53)(30 54 32 56)(33 69 35 71)(34 72 36 70)(37 75 39 73)(38 74 40 76)(57 93 59 95)(58 96 60 94)(77 118 79 120)(78 117 80 119)(81 114 83 116)(82 113 84 115)(85 126 87 128)(86 125 88 127)(89 122 91 124)(90 121 92 123)(97 133 99 135)(98 136 100 134)(101 137 103 139)(102 140 104 138)(105 141 107 143)(106 144 108 142)
G:=sub<Sym(144)| (1,59,7)(2,60,8)(3,57,5)(4,58,6)(9,30,27)(10,31,28)(11,32,25)(12,29,26)(13,39,34)(14,40,35)(15,37,36)(16,38,33)(17,144,123)(18,141,124)(19,142,121)(20,143,122)(21,113,118)(22,114,119)(23,115,120)(24,116,117)(41,46,95)(42,47,96)(43,48,93)(44,45,94)(49,75,70)(50,76,71)(51,73,72)(52,74,69)(53,66,62)(54,67,63)(55,68,64)(56,65,61)(77,131,82)(78,132,83)(79,129,84)(80,130,81)(85,103,99)(86,104,100)(87,101,97)(88,102,98)(89,112,107)(90,109,108)(91,110,105)(92,111,106)(125,138,134)(126,139,135)(127,140,136)(128,137,133), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,22,137)(18,23,138)(19,24,139)(20,21,140)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,64,71)(42,61,72)(43,62,69)(44,63,70)(45,54,49)(46,55,50)(47,56,51)(48,53,52)(65,73,96)(66,74,93)(67,75,94)(68,76,95)(77,88,89)(78,85,90)(79,86,91)(80,87,92)(81,97,106)(82,98,107)(83,99,108)(84,100,105)(101,111,130)(102,112,131)(103,109,132)(104,110,129)(113,136,143)(114,133,144)(115,134,141)(116,135,142)(117,126,121)(118,127,122)(119,128,123)(120,125,124), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,137,22)(18,138,23)(19,139,24)(20,140,21)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,64,71)(42,61,72)(43,62,69)(44,63,70)(45,54,49)(46,55,50)(47,56,51)(48,53,52)(65,73,96)(66,74,93)(67,75,94)(68,76,95)(77,89,88)(78,90,85)(79,91,86)(80,92,87)(81,106,97)(82,107,98)(83,108,99)(84,105,100)(101,130,111)(102,131,112)(103,132,109)(104,129,110)(113,143,136)(114,144,133)(115,141,134)(116,142,135)(117,121,126)(118,122,127)(119,123,128)(120,124,125), (1,77)(2,78)(3,79)(4,80)(5,84)(6,81)(7,82)(8,83)(9,99)(10,100)(11,97)(12,98)(13,92)(14,89)(15,90)(16,91)(17,73)(18,74)(19,75)(20,76)(21,95)(22,96)(23,93)(24,94)(25,101)(26,102)(27,103)(28,104)(29,88)(30,85)(31,86)(32,87)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,46,3,48)(2,45,4,47)(5,43,7,41)(6,42,8,44)(9,63,11,61)(10,62,12,64)(13,51,15,49)(14,50,16,52)(17,109,19,111)(18,112,20,110)(21,129,23,131)(22,132,24,130)(25,65,27,67)(26,68,28,66)(29,55,31,53)(30,54,32,56)(33,69,35,71)(34,72,36,70)(37,75,39,73)(38,74,40,76)(57,93,59,95)(58,96,60,94)(77,118,79,120)(78,117,80,119)(81,114,83,116)(82,113,84,115)(85,126,87,128)(86,125,88,127)(89,122,91,124)(90,121,92,123)(97,133,99,135)(98,136,100,134)(101,137,103,139)(102,140,104,138)(105,141,107,143)(106,144,108,142)>;
G:=Group( (1,59,7)(2,60,8)(3,57,5)(4,58,6)(9,30,27)(10,31,28)(11,32,25)(12,29,26)(13,39,34)(14,40,35)(15,37,36)(16,38,33)(17,144,123)(18,141,124)(19,142,121)(20,143,122)(21,113,118)(22,114,119)(23,115,120)(24,116,117)(41,46,95)(42,47,96)(43,48,93)(44,45,94)(49,75,70)(50,76,71)(51,73,72)(52,74,69)(53,66,62)(54,67,63)(55,68,64)(56,65,61)(77,131,82)(78,132,83)(79,129,84)(80,130,81)(85,103,99)(86,104,100)(87,101,97)(88,102,98)(89,112,107)(90,109,108)(91,110,105)(92,111,106)(125,138,134)(126,139,135)(127,140,136)(128,137,133), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,22,137)(18,23,138)(19,24,139)(20,21,140)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,64,71)(42,61,72)(43,62,69)(44,63,70)(45,54,49)(46,55,50)(47,56,51)(48,53,52)(65,73,96)(66,74,93)(67,75,94)(68,76,95)(77,88,89)(78,85,90)(79,86,91)(80,87,92)(81,97,106)(82,98,107)(83,99,108)(84,100,105)(101,111,130)(102,112,131)(103,109,132)(104,110,129)(113,136,143)(114,133,144)(115,134,141)(116,135,142)(117,126,121)(118,127,122)(119,128,123)(120,125,124), (1,29,14)(2,30,15)(3,31,16)(4,32,13)(5,10,33)(6,11,34)(7,12,35)(8,9,36)(17,137,22)(18,138,23)(19,139,24)(20,140,21)(25,39,58)(26,40,59)(27,37,60)(28,38,57)(41,64,71)(42,61,72)(43,62,69)(44,63,70)(45,54,49)(46,55,50)(47,56,51)(48,53,52)(65,73,96)(66,74,93)(67,75,94)(68,76,95)(77,89,88)(78,90,85)(79,91,86)(80,92,87)(81,106,97)(82,107,98)(83,108,99)(84,105,100)(101,130,111)(102,131,112)(103,132,109)(104,129,110)(113,143,136)(114,144,133)(115,141,134)(116,142,135)(117,121,126)(118,122,127)(119,123,128)(120,124,125), (1,77)(2,78)(3,79)(4,80)(5,84)(6,81)(7,82)(8,83)(9,99)(10,100)(11,97)(12,98)(13,92)(14,89)(15,90)(16,91)(17,73)(18,74)(19,75)(20,76)(21,95)(22,96)(23,93)(24,94)(25,101)(26,102)(27,103)(28,104)(29,88)(30,85)(31,86)(32,87)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,46,3,48)(2,45,4,47)(5,43,7,41)(6,42,8,44)(9,63,11,61)(10,62,12,64)(13,51,15,49)(14,50,16,52)(17,109,19,111)(18,112,20,110)(21,129,23,131)(22,132,24,130)(25,65,27,67)(26,68,28,66)(29,55,31,53)(30,54,32,56)(33,69,35,71)(34,72,36,70)(37,75,39,73)(38,74,40,76)(57,93,59,95)(58,96,60,94)(77,118,79,120)(78,117,80,119)(81,114,83,116)(82,113,84,115)(85,126,87,128)(86,125,88,127)(89,122,91,124)(90,121,92,123)(97,133,99,135)(98,136,100,134)(101,137,103,139)(102,140,104,138)(105,141,107,143)(106,144,108,142) );
G=PermutationGroup([[(1,59,7),(2,60,8),(3,57,5),(4,58,6),(9,30,27),(10,31,28),(11,32,25),(12,29,26),(13,39,34),(14,40,35),(15,37,36),(16,38,33),(17,144,123),(18,141,124),(19,142,121),(20,143,122),(21,113,118),(22,114,119),(23,115,120),(24,116,117),(41,46,95),(42,47,96),(43,48,93),(44,45,94),(49,75,70),(50,76,71),(51,73,72),(52,74,69),(53,66,62),(54,67,63),(55,68,64),(56,65,61),(77,131,82),(78,132,83),(79,129,84),(80,130,81),(85,103,99),(86,104,100),(87,101,97),(88,102,98),(89,112,107),(90,109,108),(91,110,105),(92,111,106),(125,138,134),(126,139,135),(127,140,136),(128,137,133)], [(1,29,14),(2,30,15),(3,31,16),(4,32,13),(5,10,33),(6,11,34),(7,12,35),(8,9,36),(17,22,137),(18,23,138),(19,24,139),(20,21,140),(25,39,58),(26,40,59),(27,37,60),(28,38,57),(41,64,71),(42,61,72),(43,62,69),(44,63,70),(45,54,49),(46,55,50),(47,56,51),(48,53,52),(65,73,96),(66,74,93),(67,75,94),(68,76,95),(77,88,89),(78,85,90),(79,86,91),(80,87,92),(81,97,106),(82,98,107),(83,99,108),(84,100,105),(101,111,130),(102,112,131),(103,109,132),(104,110,129),(113,136,143),(114,133,144),(115,134,141),(116,135,142),(117,126,121),(118,127,122),(119,128,123),(120,125,124)], [(1,29,14),(2,30,15),(3,31,16),(4,32,13),(5,10,33),(6,11,34),(7,12,35),(8,9,36),(17,137,22),(18,138,23),(19,139,24),(20,140,21),(25,39,58),(26,40,59),(27,37,60),(28,38,57),(41,64,71),(42,61,72),(43,62,69),(44,63,70),(45,54,49),(46,55,50),(47,56,51),(48,53,52),(65,73,96),(66,74,93),(67,75,94),(68,76,95),(77,89,88),(78,90,85),(79,91,86),(80,92,87),(81,106,97),(82,107,98),(83,108,99),(84,105,100),(101,130,111),(102,131,112),(103,132,109),(104,129,110),(113,143,136),(114,144,133),(115,141,134),(116,142,135),(117,121,126),(118,122,127),(119,123,128),(120,124,125)], [(1,77),(2,78),(3,79),(4,80),(5,84),(6,81),(7,82),(8,83),(9,99),(10,100),(11,97),(12,98),(13,92),(14,89),(15,90),(16,91),(17,73),(18,74),(19,75),(20,76),(21,95),(22,96),(23,93),(24,94),(25,101),(26,102),(27,103),(28,104),(29,88),(30,85),(31,86),(32,87),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(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)], [(1,46,3,48),(2,45,4,47),(5,43,7,41),(6,42,8,44),(9,63,11,61),(10,62,12,64),(13,51,15,49),(14,50,16,52),(17,109,19,111),(18,112,20,110),(21,129,23,131),(22,132,24,130),(25,65,27,67),(26,68,28,66),(29,55,31,53),(30,54,32,56),(33,69,35,71),(34,72,36,70),(37,75,39,73),(38,74,40,76),(57,93,59,95),(58,96,60,94),(77,118,79,120),(78,117,80,119),(81,114,83,116),(82,113,84,115),(85,126,87,128),(86,125,88,127),(89,122,91,124),(90,121,92,123),(97,133,99,135),(98,136,100,134),(101,137,103,139),(102,140,104,138),(105,141,107,143),(106,144,108,142)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6AG | 12A | ··· | 12X | 12Y | ··· | 12AY | 12AZ | ··· | 12BW |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | C3×S3 | C3×Q8 | S3×C6 | S3×Q8 | C3×S3×Q8 |
| kernel | S3×Q8×C32 | C32×Dic6 | S3×C3×C12 | Q8×C33 | C3×S3×Q8 | C3×Dic6 | S3×C12 | Q8×C32 | Q8×C32 | S3×C32 | C3×C12 | C3×Q8 | C3×S3 | C12 | C32 | C3 |
| # reps | 1 | 3 | 3 | 1 | 8 | 24 | 24 | 8 | 1 | 2 | 3 | 8 | 16 | 24 | 1 | 8 |
Matrix representation of S3×Q8×C32 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 4 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 3 | 2 |
| 0 | 0 | 0 | 9 | 10 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 8 | 5 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 5 | 0 | 0 |
| 0 | 7 | 9 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,4,0,0,0,0,3],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,3,9,0,0,0,2,10],[12,0,0,0,0,0,8,8,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,4,7,0,0,0,5,9,0,0,0,0,0,12,0,0,0,0,0,12] >;
S3×Q8×C32 in GAP, Magma, Sage, TeX
S_3\times Q_8\times C_3^2
% in TeX
G:=Group("S3xQ8xC3^2"); // GroupNames label
G:=SmallGroup(432,706);
// by ID
G=gap.SmallGroup(432,706);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,512,807,394,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^4=1,f^2=e^2,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,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations