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## G = S3×C3×C24order 432 = 24·33

### Direct product of C3×C24 and S3

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 280 in 164 conjugacy classes, 90 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×C6, C3×C6, C3×C6, C3⋊C8, C24, C24, C24, C4×S3, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, S3×C8, C2×C24, S3×C32, C32×C6, C3×C3⋊C8, C3×C24, C3×C24, C3×C24, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, S3×C24, C6×C24, C32×C3⋊C8, C32×C24, S3×C3×C12, S3×C3×C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C32, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C24, C4×S3, C2×C12, C3×C12, S3×C6, C62, S3×C8, C2×C24, S3×C32, C3×C24, S3×C12, C6×C12, S3×C3×C6, S3×C24, C6×C24, S3×C3×C12, S3×C3×C24

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

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

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

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

216 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 6A ··· 6H 6I ··· 6Q 6R ··· 6AG 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12P 12Q ··· 12AH 12AI ··· 12AX 24A ··· 24AF 24AG ··· 24BP 24BQ ··· 24CV order 1 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 3 3 1 ··· 1 2 ··· 2 1 1 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 1 1 1 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

216 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 S3×C3×C24 C32×C3⋊C8 C32×C24 S3×C3×C12 S3×C24 C32×Dic3 S3×C3×C6 C3×C3⋊C8 C3×C24 S3×C12 S3×C32 C3×Dic3 S3×C6 C3×S3 C3×C24 C3×C12 C24 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 8 2 2 8 8 8 8 16 16 64 1 1 8 2 8 4 16 32

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

 64 0 0 0 0 1 0 0 0 0 64 0 0 0 0 64
,
 52 0 0 0 0 8 0 0 0 0 24 0 0 0 0 24
,
 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 64
,
 72 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [64,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[52,0,0,0,0,8,0,0,0,0,24,0,0,0,0,24],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[72,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

S3×C3×C24 in GAP, Magma, Sage, TeX

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

G:=Group("S3xC3xC24");
// GroupNames label

G:=SmallGroup(432,464);
// by ID

G=gap.SmallGroup(432,464);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,260,102,14118]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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