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## G = S3×Q8×C32order 432 = 24·33

### Direct product of C32, S3 and Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×Q8×C32
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C3×C12 — S3×Q8×C32
 Lower central C3 — C6 — S3×Q8×C32
 Upper central C1 — C3×C6 — Q8×C32

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

Smallest permutation representation of 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

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