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## G = C3×C24⋊2S3order 432 = 24·33

### Direct product of C3 and C24⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C24⋊2S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C12⋊S3 — C3×C24⋊2S3
 Lower central C32 — C3×C6 — C3×C12 — C3×C24⋊2S3
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×C242S3
G = < a,b,c,d | a3=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b11, dcd=c-1 >

Subgroups: 596 in 152 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C24, C24, C24, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, C3×SD16, C3×C3⋊S3, C32×C6, C3×C24, C3×C24, C3×C24, C3×Dic6, C3×D12, C324Q8, C12⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×C24⋊C2, C242S3, C32×C24, C3×C324Q8, C3×C12⋊S3, C3×C242S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊S3, D12, C3×D4, S3×C6, C2×C3⋊S3, C24⋊C2, C3×SD16, C3×C3⋊S3, C3×D12, C12⋊S3, C6×C3⋊S3, C3×C24⋊C2, C242S3, C3×C12⋊S3, C3×C242S3

Smallest permutation representation of C3×C242S3
On 144 points
Generators in S144
(1 115 45)(2 116 46)(3 117 47)(4 118 48)(5 119 25)(6 120 26)(7 97 27)(8 98 28)(9 99 29)(10 100 30)(11 101 31)(12 102 32)(13 103 33)(14 104 34)(15 105 35)(16 106 36)(17 107 37)(18 108 38)(19 109 39)(20 110 40)(21 111 41)(22 112 42)(23 113 43)(24 114 44)(49 92 121)(50 93 122)(51 94 123)(52 95 124)(53 96 125)(54 73 126)(55 74 127)(56 75 128)(57 76 129)(58 77 130)(59 78 131)(60 79 132)(61 80 133)(62 81 134)(63 82 135)(64 83 136)(65 84 137)(66 85 138)(67 86 139)(68 87 140)(69 88 141)(70 89 142)(71 90 143)(72 91 144)
(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)
(1 107 29)(2 108 30)(3 109 31)(4 110 32)(5 111 33)(6 112 34)(7 113 35)(8 114 36)(9 115 37)(10 116 38)(11 117 39)(12 118 40)(13 119 41)(14 120 42)(15 97 43)(16 98 44)(17 99 45)(18 100 46)(19 101 47)(20 102 48)(21 103 25)(22 104 26)(23 105 27)(24 106 28)(49 137 76)(50 138 77)(51 139 78)(52 140 79)(53 141 80)(54 142 81)(55 143 82)(56 144 83)(57 121 84)(58 122 85)(59 123 86)(60 124 87)(61 125 88)(62 126 89)(63 127 90)(64 128 91)(65 129 92)(66 130 93)(67 131 94)(68 132 95)(69 133 96)(70 134 73)(71 135 74)(72 136 75)
(1 82)(2 93)(3 80)(4 91)(5 78)(6 89)(7 76)(8 87)(9 74)(10 85)(11 96)(12 83)(13 94)(14 81)(15 92)(16 79)(17 90)(18 77)(19 88)(20 75)(21 86)(22 73)(23 84)(24 95)(25 59)(26 70)(27 57)(28 68)(29 55)(30 66)(31 53)(32 64)(33 51)(34 62)(35 49)(36 60)(37 71)(38 58)(39 69)(40 56)(41 67)(42 54)(43 65)(44 52)(45 63)(46 50)(47 61)(48 72)(97 129)(98 140)(99 127)(100 138)(101 125)(102 136)(103 123)(104 134)(105 121)(106 132)(107 143)(108 130)(109 141)(110 128)(111 139)(112 126)(113 137)(114 124)(115 135)(116 122)(117 133)(118 144)(119 131)(120 142)

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

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

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

117 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6N 6O 6P 8A 8B 12A ··· 12Z 12AA 12AB 24A ··· 24AZ order 1 2 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 6 6 8 8 12 ··· 12 12 12 24 ··· 24 size 1 1 36 1 1 2 ··· 2 2 36 1 1 2 ··· 2 36 36 2 2 2 ··· 2 36 36 2 ··· 2

117 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 SD16 C3×S3 D12 C3×D4 S3×C6 C24⋊C2 C3×SD16 C3×D12 C3×C24⋊C2 kernel C3×C24⋊2S3 C32×C24 C3×C32⋊4Q8 C3×C12⋊S3 C24⋊2S3 C3×C24 C32⋊4Q8 C12⋊S3 C3×C24 C32×C6 C3×C12 C33 C24 C3×C6 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 1 1 2 2 2 2 4 1 4 2 8 8 2 8 16 4 16 32

Matrix representation of C3×C242S3 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 56 0 0 0 0 43
,
 8 0 0 0 0 64 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],[1,0,0,0,0,1,0,0,0,0,56,0,0,0,0,43],[8,0,0,0,0,64,0,0,0,0,8,0,0,0,0,64],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C242S3 in GAP, Magma, Sage, TeX

C_3\times C_{24}\rtimes_2S_3
% in TeX

G:=Group("C3xC24:2S3");
// GroupNames label

G:=SmallGroup(432,482);
// by ID

G=gap.SmallGroup(432,482);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,4037,14118]);
// 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^11,d*c*d=c^-1>;
// generators/relations

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