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## G = C3×C24⋊S3order 432 = 24·33

### Direct product of C3 and C24⋊S3

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C24⋊S3
G = < a,b,c,d | a3=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 420 in 152 conjugacy classes, 58 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, M4(2), 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, C8⋊S3, C3×M4(2), 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, C3×C8⋊S3, C24⋊S3, C3×C324C8, C32×C24, C12×C3⋊S3, C3×C24⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C4×S3, C2×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C3×M4(2), C3×C3⋊S3, S3×C12, C4×C3⋊S3, C6×C3⋊S3, C3×C8⋊S3, C24⋊S3, C12×C3⋊S3, C3×C24⋊S3

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

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

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

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

126 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3N 4A 4B 4C 6A 6B 6C ··· 6N 6O 6P 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12AB 12AC 12AD 24A ··· 24AZ 24BA 24BB 24BC 24BD order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 ··· 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 24 ··· 24 24 24 24 24 size 1 1 18 1 1 2 ··· 2 1 1 18 1 1 2 ··· 2 18 18 2 2 18 18 1 1 1 1 2 ··· 2 18 18 2 ··· 2 18 18 18 18

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D6 M4(2) C3×S3 C4×S3 S3×C6 C8⋊S3 C3×M4(2) S3×C12 C3×C8⋊S3 kernel C3×C24⋊S3 C3×C32⋊4C8 C32×C24 C12×C3⋊S3 C24⋊S3 C3×C3⋊Dic3 C6×C3⋊S3 C32⋊4C8 C3×C24 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3×C24 C3×C12 C33 C24 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 1 1 2 2 2 2 2 2 4 4 4 4 2 8 8 8 16 4 16 32

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

 8 0 0 0 0 8 0 0 0 0 64 0 0 0 0 64
,
 43 0 0 0 0 21 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 64
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[43,0,0,0,0,21,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C24⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC24:S3");
// GroupNames label

G:=SmallGroup(432,481);
// by ID

G=gap.SmallGroup(432,481);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^24=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^5,d*c*d=c^-1>;
// generators/relations

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