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## G = S3×C32⋊4C8order 432 = 24·33

### Direct product of S3 and C32⋊4C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×C32⋊4C8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — S3×C3×C12 — S3×C32⋊4C8
 Lower central C33 — S3×C32⋊4C8
 Upper central C1 — C4

Generators and relations for S3×C324C8
G = < a,b,c,d,e | a3=b2=c3=d3=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 520 in 164 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, S3×C8, C2×C3⋊C8, S3×C32, C32×C6, C3×C3⋊C8, C324C8, C324C8, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, S3×C3⋊C8, C2×C324C8, C3×C324C8, C337C8, S3×C3×C12, S3×C324C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊S3, C3⋊C8, C4×S3, C2×Dic3, C3⋊Dic3, S32, C2×C3⋊S3, S3×C8, C2×C3⋊C8, C324C8, S3×Dic3, C2×C3⋊Dic3, S3×C3⋊S3, S3×C3⋊C8, C2×C324C8, S3×C3⋊Dic3, S3×C324C8

Smallest permutation representation of S3×C324C8
On 144 points
Generators in S144
(1 120 95)(2 113 96)(3 114 89)(4 115 90)(5 116 91)(6 117 92)(7 118 93)(8 119 94)(9 40 44)(10 33 45)(11 34 46)(12 35 47)(13 36 48)(14 37 41)(15 38 42)(16 39 43)(17 32 55)(18 25 56)(19 26 49)(20 27 50)(21 28 51)(22 29 52)(23 30 53)(24 31 54)(57 110 142)(58 111 143)(59 112 144)(60 105 137)(61 106 138)(62 107 139)(63 108 140)(64 109 141)(65 122 103)(66 123 104)(67 124 97)(68 125 98)(69 126 99)(70 127 100)(71 128 101)(72 121 102)(73 88 131)(74 81 132)(75 82 133)(76 83 134)(77 84 135)(78 85 136)(79 86 129)(80 87 130)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 80)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 96)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 113)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 129)(40 130)(41 84)(42 85)(43 86)(44 87)(45 88)(46 81)(47 82)(48 83)(57 100)(58 101)(59 102)(60 103)(61 104)(62 97)(63 98)(64 99)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 121)
(1 125 75)(2 76 126)(3 127 77)(4 78 128)(5 121 79)(6 80 122)(7 123 73)(8 74 124)(9 105 51)(10 52 106)(11 107 53)(12 54 108)(13 109 55)(14 56 110)(15 111 49)(16 50 112)(17 36 141)(18 142 37)(19 38 143)(20 144 39)(21 40 137)(22 138 33)(23 34 139)(24 140 35)(25 57 41)(26 42 58)(27 59 43)(28 44 60)(29 61 45)(30 46 62)(31 63 47)(32 48 64)(65 92 130)(66 131 93)(67 94 132)(68 133 95)(69 96 134)(70 135 89)(71 90 136)(72 129 91)(81 97 119)(82 120 98)(83 99 113)(84 114 100)(85 101 115)(86 116 102)(87 103 117)(88 118 104)
(1 68 82)(2 83 69)(3 70 84)(4 85 71)(5 72 86)(6 87 65)(7 66 88)(8 81 67)(9 60 21)(10 22 61)(11 62 23)(12 24 63)(13 64 17)(14 18 57)(15 58 19)(16 20 59)(25 110 37)(26 38 111)(27 112 39)(28 40 105)(29 106 33)(30 34 107)(31 108 35)(32 36 109)(41 56 142)(42 143 49)(43 50 144)(44 137 51)(45 52 138)(46 139 53)(47 54 140)(48 141 55)(73 93 104)(74 97 94)(75 95 98)(76 99 96)(77 89 100)(78 101 90)(79 91 102)(80 103 92)(113 134 126)(114 127 135)(115 136 128)(116 121 129)(117 130 122)(118 123 131)(119 132 124)(120 125 133)
(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)

G:=sub<Sym(144)| (1,120,95)(2,113,96)(3,114,89)(4,115,90)(5,116,91)(6,117,92)(7,118,93)(8,119,94)(9,40,44)(10,33,45)(11,34,46)(12,35,47)(13,36,48)(14,37,41)(15,38,42)(16,39,43)(17,32,55)(18,25,56)(19,26,49)(20,27,50)(21,28,51)(22,29,52)(23,30,53)(24,31,54)(57,110,142)(58,111,143)(59,112,144)(60,105,137)(61,106,138)(62,107,139)(63,108,140)(64,109,141)(65,122,103)(66,123,104)(67,124,97)(68,125,98)(69,126,99)(70,127,100)(71,128,101)(72,121,102)(73,88,131)(74,81,132)(75,82,133)(76,83,134)(77,84,135)(78,85,136)(79,86,129)(80,87,130), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,80)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,96)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,113)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,129)(40,130)(41,84)(42,85)(43,86)(44,87)(45,88)(46,81)(47,82)(48,83)(57,100)(58,101)(59,102)(60,103)(61,104)(62,97)(63,98)(64,99)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,121), (1,125,75)(2,76,126)(3,127,77)(4,78,128)(5,121,79)(6,80,122)(7,123,73)(8,74,124)(9,105,51)(10,52,106)(11,107,53)(12,54,108)(13,109,55)(14,56,110)(15,111,49)(16,50,112)(17,36,141)(18,142,37)(19,38,143)(20,144,39)(21,40,137)(22,138,33)(23,34,139)(24,140,35)(25,57,41)(26,42,58)(27,59,43)(28,44,60)(29,61,45)(30,46,62)(31,63,47)(32,48,64)(65,92,130)(66,131,93)(67,94,132)(68,133,95)(69,96,134)(70,135,89)(71,90,136)(72,129,91)(81,97,119)(82,120,98)(83,99,113)(84,114,100)(85,101,115)(86,116,102)(87,103,117)(88,118,104), (1,68,82)(2,83,69)(3,70,84)(4,85,71)(5,72,86)(6,87,65)(7,66,88)(8,81,67)(9,60,21)(10,22,61)(11,62,23)(12,24,63)(13,64,17)(14,18,57)(15,58,19)(16,20,59)(25,110,37)(26,38,111)(27,112,39)(28,40,105)(29,106,33)(30,34,107)(31,108,35)(32,36,109)(41,56,142)(42,143,49)(43,50,144)(44,137,51)(45,52,138)(46,139,53)(47,54,140)(48,141,55)(73,93,104)(74,97,94)(75,95,98)(76,99,96)(77,89,100)(78,101,90)(79,91,102)(80,103,92)(113,134,126)(114,127,135)(115,136,128)(116,121,129)(117,130,122)(118,123,131)(119,132,124)(120,125,133), (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)>;

G:=Group( (1,120,95)(2,113,96)(3,114,89)(4,115,90)(5,116,91)(6,117,92)(7,118,93)(8,119,94)(9,40,44)(10,33,45)(11,34,46)(12,35,47)(13,36,48)(14,37,41)(15,38,42)(16,39,43)(17,32,55)(18,25,56)(19,26,49)(20,27,50)(21,28,51)(22,29,52)(23,30,53)(24,31,54)(57,110,142)(58,111,143)(59,112,144)(60,105,137)(61,106,138)(62,107,139)(63,108,140)(64,109,141)(65,122,103)(66,123,104)(67,124,97)(68,125,98)(69,126,99)(70,127,100)(71,128,101)(72,121,102)(73,88,131)(74,81,132)(75,82,133)(76,83,134)(77,84,135)(78,85,136)(79,86,129)(80,87,130), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,80)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,96)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,113)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,129)(40,130)(41,84)(42,85)(43,86)(44,87)(45,88)(46,81)(47,82)(48,83)(57,100)(58,101)(59,102)(60,103)(61,104)(62,97)(63,98)(64,99)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,121), (1,125,75)(2,76,126)(3,127,77)(4,78,128)(5,121,79)(6,80,122)(7,123,73)(8,74,124)(9,105,51)(10,52,106)(11,107,53)(12,54,108)(13,109,55)(14,56,110)(15,111,49)(16,50,112)(17,36,141)(18,142,37)(19,38,143)(20,144,39)(21,40,137)(22,138,33)(23,34,139)(24,140,35)(25,57,41)(26,42,58)(27,59,43)(28,44,60)(29,61,45)(30,46,62)(31,63,47)(32,48,64)(65,92,130)(66,131,93)(67,94,132)(68,133,95)(69,96,134)(70,135,89)(71,90,136)(72,129,91)(81,97,119)(82,120,98)(83,99,113)(84,114,100)(85,101,115)(86,116,102)(87,103,117)(88,118,104), (1,68,82)(2,83,69)(3,70,84)(4,85,71)(5,72,86)(6,87,65)(7,66,88)(8,81,67)(9,60,21)(10,22,61)(11,62,23)(12,24,63)(13,64,17)(14,18,57)(15,58,19)(16,20,59)(25,110,37)(26,38,111)(27,112,39)(28,40,105)(29,106,33)(30,34,107)(31,108,35)(32,36,109)(41,56,142)(42,143,49)(43,50,144)(44,137,51)(45,52,138)(46,139,53)(47,54,140)(48,141,55)(73,93,104)(74,97,94)(75,95,98)(76,99,96)(77,89,100)(78,101,90)(79,91,102)(80,103,92)(113,134,126)(114,127,135)(115,136,128)(116,121,129)(117,130,122)(118,123,131)(119,132,124)(120,125,133), (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) );

G=PermutationGroup([[(1,120,95),(2,113,96),(3,114,89),(4,115,90),(5,116,91),(6,117,92),(7,118,93),(8,119,94),(9,40,44),(10,33,45),(11,34,46),(12,35,47),(13,36,48),(14,37,41),(15,38,42),(16,39,43),(17,32,55),(18,25,56),(19,26,49),(20,27,50),(21,28,51),(22,29,52),(23,30,53),(24,31,54),(57,110,142),(58,111,143),(59,112,144),(60,105,137),(61,106,138),(62,107,139),(63,108,140),(64,109,141),(65,122,103),(66,123,104),(67,124,97),(68,125,98),(69,126,99),(70,127,100),(71,128,101),(72,121,102),(73,88,131),(74,81,132),(75,82,133),(76,83,134),(77,84,135),(78,85,136),(79,86,129),(80,87,130)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,80),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,96),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,113),(33,131),(34,132),(35,133),(36,134),(37,135),(38,136),(39,129),(40,130),(41,84),(42,85),(43,86),(44,87),(45,88),(46,81),(47,82),(48,83),(57,100),(58,101),(59,102),(60,103),(61,104),(62,97),(63,98),(64,99),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144),(105,122),(106,123),(107,124),(108,125),(109,126),(110,127),(111,128),(112,121)], [(1,125,75),(2,76,126),(3,127,77),(4,78,128),(5,121,79),(6,80,122),(7,123,73),(8,74,124),(9,105,51),(10,52,106),(11,107,53),(12,54,108),(13,109,55),(14,56,110),(15,111,49),(16,50,112),(17,36,141),(18,142,37),(19,38,143),(20,144,39),(21,40,137),(22,138,33),(23,34,139),(24,140,35),(25,57,41),(26,42,58),(27,59,43),(28,44,60),(29,61,45),(30,46,62),(31,63,47),(32,48,64),(65,92,130),(66,131,93),(67,94,132),(68,133,95),(69,96,134),(70,135,89),(71,90,136),(72,129,91),(81,97,119),(82,120,98),(83,99,113),(84,114,100),(85,101,115),(86,116,102),(87,103,117),(88,118,104)], [(1,68,82),(2,83,69),(3,70,84),(4,85,71),(5,72,86),(6,87,65),(7,66,88),(8,81,67),(9,60,21),(10,22,61),(11,62,23),(12,24,63),(13,64,17),(14,18,57),(15,58,19),(16,20,59),(25,110,37),(26,38,111),(27,112,39),(28,40,105),(29,106,33),(30,34,107),(31,108,35),(32,36,109),(41,56,142),(42,143,49),(43,50,144),(44,137,51),(45,52,138),(46,139,53),(47,54,140),(48,141,55),(73,93,104),(74,97,94),(75,95,98),(76,99,96),(77,89,100),(78,101,90),(79,91,102),(80,103,92),(113,134,126),(114,127,135),(115,136,128),(116,121,129),(117,130,122),(118,123,131),(119,132,124),(120,125,133)], [(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)]])

72 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12J 12K ··· 12R 12S ··· 12Z 24A 24B 24C 24D order 1 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 3 3 2 ··· 2 4 4 4 4 1 1 3 3 2 ··· 2 4 4 4 4 6 ··· 6 9 9 9 9 27 27 27 27 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C8 S3 S3 Dic3 D6 Dic3 C3⋊C8 C4×S3 S3×C8 S32 S3×Dic3 S3×C3⋊C8 kernel S3×C32⋊4C8 C3×C32⋊4C8 C33⋊7C8 S3×C3×C12 C32×Dic3 S3×C3×C6 S3×C32 C32⋊4C8 S3×C12 C3×Dic3 C3×C12 S3×C6 C3×S3 C3×C6 C32 C12 C6 C3 # reps 1 1 1 1 2 2 8 1 4 4 5 4 16 2 4 4 4 8

Matrix representation of S3×C324C8 in GL6(𝔽73)

 1 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 72 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 72 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 1 0
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 1 0 0 0 0 72 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
,
 44 54 0 0 0 0 25 29 0 0 0 0 0 0 0 51 0 0 0 0 51 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[44,25,0,0,0,0,54,29,0,0,0,0,0,0,0,51,0,0,0,0,51,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S3×C324C8 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes_4C_8
% in TeX

G:=Group("S3xC3^2:4C8");
// GroupNames label

G:=SmallGroup(432,430);
// by ID

G=gap.SmallGroup(432,430);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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