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## G = C2×C24⋊S3order 288 = 25·32

### Direct product of C2 and C24⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C24⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — C2×C4×C3⋊S3 — C2×C24⋊S3
 Lower central C32 — C3×C6 — C2×C24⋊S3
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C2×C24⋊S3
G = < a,b,c,d | a2=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 660 in 204 conjugacy classes, 85 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C324C8, C3×C24, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C8⋊S3, C24⋊S3, C2×C324C8, C6×C24, C2×C4×C3⋊S3, C2×C24⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C3⋊S3, C4×S3, C22×S3, C2×M4(2), C2×C3⋊S3, C8⋊S3, S3×C2×C4, C4×C3⋊S3, C22×C3⋊S3, C2×C8⋊S3, C24⋊S3, C2×C4×C3⋊S3, C2×C24⋊S3

Smallest permutation representation of C2×C24⋊S3
On 144 points
Generators in S144
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 73)(22 74)(23 75)(24 76)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(103 140)(104 141)(105 142)(106 143)(107 144)(108 121)(109 122)(110 123)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 133)
(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)
(1 48 141)(2 25 142)(3 26 143)(4 27 144)(5 28 121)(6 29 122)(7 30 123)(8 31 124)(9 32 125)(10 33 126)(11 34 127)(12 35 128)(13 36 129)(14 37 130)(15 38 131)(16 39 132)(17 40 133)(18 41 134)(19 42 135)(20 43 136)(21 44 137)(22 45 138)(23 46 139)(24 47 140)(49 120 93)(50 97 94)(51 98 95)(52 99 96)(53 100 73)(54 101 74)(55 102 75)(56 103 76)(57 104 77)(58 105 78)(59 106 79)(60 107 80)(61 108 81)(62 109 82)(63 110 83)(64 111 84)(65 112 85)(66 113 86)(67 114 87)(68 115 88)(69 116 89)(70 117 90)(71 118 91)(72 119 92)
(1 77)(2 82)(3 87)(4 92)(5 73)(6 78)(7 83)(8 88)(9 93)(10 74)(11 79)(12 84)(13 89)(14 94)(15 75)(16 80)(17 85)(18 90)(19 95)(20 76)(21 81)(22 86)(23 91)(24 96)(25 109)(26 114)(27 119)(28 100)(29 105)(30 110)(31 115)(32 120)(33 101)(34 106)(35 111)(36 116)(37 97)(38 102)(39 107)(40 112)(41 117)(42 98)(43 103)(44 108)(45 113)(46 118)(47 99)(48 104)(49 125)(50 130)(51 135)(52 140)(53 121)(54 126)(55 131)(56 136)(57 141)(58 122)(59 127)(60 132)(61 137)(62 142)(63 123)(64 128)(65 133)(66 138)(67 143)(68 124)(69 129)(70 134)(71 139)(72 144)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6L 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 18 18 2 2 2 2 1 1 1 1 18 18 2 ··· 2 2 2 2 2 18 18 18 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 M4(2) C4×S3 C4×S3 C8⋊S3 kernel C2×C24⋊S3 C24⋊S3 C2×C32⋊4C8 C6×C24 C2×C4×C3⋊S3 C4×C3⋊S3 C2×C3⋊Dic3 C22×C3⋊S3 C2×C24 C24 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 4 1 1 1 4 2 2 4 8 4 4 8 8 32

Matrix representation of C2×C24⋊S3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 46 0 0 0 0 46 0 0 0 0 0 24 0 0 65 49
,
 72 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 1 1 0 0 0 0 1 0 0 0 72 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,0,46,0,0,0,0,0,65,0,0,24,49],[72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[72,1,0,0,0,1,0,0,0,0,1,72,0,0,0,72] >;

C2×C24⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_{24}\rtimes S_3
% in TeX

G:=Group("C2xC24:S3");
// GroupNames label

G:=SmallGroup(288,757);
// by ID

G=gap.SmallGroup(288,757);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,58,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=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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