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## G = C3⋊S3×C24order 432 = 24·33

### Direct product of C24 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C12×C3⋊S3 — C3⋊S3×C24
 Lower central C32 — C3⋊S3×C24
 Upper central C1 — C24

Generators and relations for C3⋊S3×C24
G = < a,b,c,d | a24=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 420 in 164 conjugacy classes, 62 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C24, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C24, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C324C8, C3×C24, C3×C24, C3×C24, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, S3×C24, C8×C3⋊S3, C3×C324C8, C32×C24, C12×C3⋊S3, C3⋊S3×C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C24, C4×S3, C2×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C24, C3×C3⋊S3, S3×C12, C4×C3⋊S3, C6×C3⋊S3, S3×C24, C8×C3⋊S3, C12×C3⋊S3, C3⋊S3×C24

Smallest permutation representation of C3⋊S3×C24
On 144 points
Generators in S144
(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)
(1 131 36)(2 132 37)(3 133 38)(4 134 39)(5 135 40)(6 136 41)(7 137 42)(8 138 43)(9 139 44)(10 140 45)(11 141 46)(12 142 47)(13 143 48)(14 144 25)(15 121 26)(16 122 27)(17 123 28)(18 124 29)(19 125 30)(20 126 31)(21 127 32)(22 128 33)(23 129 34)(24 130 35)(49 104 74)(50 105 75)(51 106 76)(52 107 77)(53 108 78)(54 109 79)(55 110 80)(56 111 81)(57 112 82)(58 113 83)(59 114 84)(60 115 85)(61 116 86)(62 117 87)(63 118 88)(64 119 89)(65 120 90)(66 97 91)(67 98 92)(68 99 93)(69 100 94)(70 101 95)(71 102 96)(72 103 73)
(1 44 123)(2 45 124)(3 46 125)(4 47 126)(5 48 127)(6 25 128)(7 26 129)(8 27 130)(9 28 131)(10 29 132)(11 30 133)(12 31 134)(13 32 135)(14 33 136)(15 34 137)(16 35 138)(17 36 139)(18 37 140)(19 38 141)(20 39 142)(21 40 143)(22 41 144)(23 42 121)(24 43 122)(49 90 112)(50 91 113)(51 92 114)(52 93 115)(53 94 116)(54 95 117)(55 96 118)(56 73 119)(57 74 120)(58 75 97)(59 76 98)(60 77 99)(61 78 100)(62 79 101)(63 80 102)(64 81 103)(65 82 104)(66 83 105)(67 84 106)(68 85 107)(69 86 108)(70 87 109)(71 88 110)(72 89 111)
(1 95)(2 96)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(97 127)(98 128)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4A 4B 4C 4D 6A 6B 6C ··· 6N 6O 6P 6Q 6R 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12AB 12AC 12AD 12AE 12AF 24A ··· 24H 24I ··· 24BD 24BE ··· 24BL order 1 2 2 2 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 12 12 12 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 9 9 1 1 2 ··· 2 1 1 9 9 1 1 2 ··· 2 9 9 9 9 1 1 1 1 9 9 9 9 1 1 1 1 2 ··· 2 9 9 9 9 1 ··· 1 2 ··· 2 9 ··· 9

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C12 C12 C24 S3 D6 C3×S3 C4×S3 S3×C6 S3×C8 S3×C12 S3×C24 kernel C3⋊S3×C24 C3×C32⋊4C8 C32×C24 C12×C3⋊S3 C8×C3⋊S3 C3×C3⋊Dic3 C6×C3⋊S3 C32⋊4C8 C3×C24 C4×C3⋊S3 C3×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 C3×C24 C3×C12 C24 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 2 2 2 2 2 2 8 4 4 16 4 4 8 8 8 16 16 32

Matrix representation of C3⋊S3×C24 in GL4(𝔽73) generated by

 43 0 0 0 0 43 0 0 0 0 65 0 0 0 0 65
,
 64 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 65 8
,
 0 72 0 0 72 0 0 0 0 0 1 7 0 0 0 72
G:=sub<GL(4,GF(73))| [43,0,0,0,0,43,0,0,0,0,65,0,0,0,0,65],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,64,65,0,0,0,8],[0,72,0,0,72,0,0,0,0,0,1,0,0,0,7,72] >;

C3⋊S3×C24 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_{24}
% in TeX

G:=Group("C3:S3xC24");
// GroupNames label

G:=SmallGroup(432,480);
// by ID

G=gap.SmallGroup(432,480);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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