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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 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×12], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C2×C8, C2×C8, M4(2) [×4], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×8], C24 [×8], C4×S3 [×16], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C2×M4(2), C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C8⋊S3 [×16], C2×C3⋊C8 [×4], C2×C24 [×4], S3×C2×C4 [×4], C324C8 [×2], C3×C24 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C8⋊S3 [×4], C24⋊S3 [×4], C2×C324C8, C6×C24, C2×C4×C3⋊S3, C2×C24⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], C23, D6 [×12], M4(2) [×2], C22×C4, C3⋊S3, C4×S3 [×8], C22×S3 [×4], C2×M4(2), C2×C3⋊S3 [×3], C8⋊S3 [×8], S3×C2×C4 [×4], C4×C3⋊S3 [×2], C22×C3⋊S3, C2×C8⋊S3 [×4], C24⋊S3 [×2], C2×C4×C3⋊S3, C2×C24⋊S3

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

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

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

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

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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