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## G = C2×C24⋊2S3order 288 = 25·32

### Direct product of C2 and C24⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C2×C24⋊2S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C2×C12⋊S3 — C2×C24⋊2S3
 Lower central C32 — C3×C6 — C3×C12 — C2×C24⋊2S3
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C242S3
G = < a,b,c,d | a2=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b11, dcd=c-1 >

Subgroups: 1012 in 204 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×12], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C32, Dic3 [×8], C12 [×8], D6 [×16], C2×C6 [×4], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C24 [×8], Dic6 [×12], D12 [×12], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×4], C2×SD16, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×4], C62, C24⋊C2 [×16], C2×C24 [×4], C2×Dic6 [×4], C2×D12 [×4], C3×C24 [×2], C324Q8 [×2], C324Q8, C12⋊S3 [×2], C12⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C24⋊C2 [×4], C242S3 [×4], C6×C24, C2×C324Q8, C2×C12⋊S3, C2×C242S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], SD16 [×2], C2×D4, C3⋊S3, D12 [×8], C22×S3 [×4], C2×SD16, C2×C3⋊S3 [×3], C24⋊C2 [×8], C2×D12 [×4], C12⋊S3 [×2], C22×C3⋊S3, C2×C24⋊C2 [×4], C242S3 [×2], C2×C12⋊S3, C2×C242S3

Smallest permutation representation of C2×C242S3
On 144 points
Generators in S144
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 141)(10 142)(11 143)(12 144)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 129)(22 130)(23 131)(24 132)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)
(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)
(1 115 69)(2 116 70)(3 117 71)(4 118 72)(5 119 49)(6 120 50)(7 97 51)(8 98 52)(9 99 53)(10 100 54)(11 101 55)(12 102 56)(13 103 57)(14 104 58)(15 105 59)(16 106 60)(17 107 61)(18 108 62)(19 109 63)(20 110 64)(21 111 65)(22 112 66)(23 113 67)(24 114 68)(25 82 131)(26 83 132)(27 84 133)(28 85 134)(29 86 135)(30 87 136)(31 88 137)(32 89 138)(33 90 139)(34 91 140)(35 92 141)(36 93 142)(37 94 143)(38 95 144)(39 96 121)(40 73 122)(41 74 123)(42 75 124)(43 76 125)(44 77 126)(45 78 127)(46 79 128)(47 80 129)(48 81 130)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 86)(26 73)(27 84)(28 95)(29 82)(30 93)(31 80)(32 91)(33 78)(34 89)(35 76)(36 87)(37 74)(38 85)(39 96)(40 83)(41 94)(42 81)(43 92)(44 79)(45 90)(46 77)(47 88)(48 75)(49 111)(50 98)(51 109)(52 120)(53 107)(54 118)(55 105)(56 116)(57 103)(58 114)(59 101)(60 112)(61 99)(62 110)(63 97)(64 108)(65 119)(66 106)(67 117)(68 104)(69 115)(70 102)(71 113)(72 100)(122 132)(123 143)(124 130)(125 141)(126 128)(127 139)(129 137)(131 135)(134 144)(136 142)(138 140)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 36 36 2 2 2 2 2 2 36 36 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 SD16 D12 D12 C24⋊C2 kernel C2×C24⋊2S3 C24⋊2S3 C6×C24 C2×C32⋊4Q8 C2×C12⋊S3 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 4 1 1 1 4 1 1 8 4 4 8 8 32

Matrix representation of C2×C242S3 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 66 7 0 0 66 59 0 0 0 0 11 25 0 0 48 36
,
 0 1 0 0 72 72 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[66,66,0,0,7,59,0,0,0,0,11,48,0,0,25,36],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[72,1,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C2×C242S3 in GAP, Magma, Sage, TeX

C_2\times C_{24}\rtimes_2S_3
% in TeX

G:=Group("C2xC24:2S3");
// GroupNames label

G:=SmallGroup(288,759);
// by ID

G=gap.SmallGroup(288,759);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,58,675,80,2693,9414]);
// 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^11,d*c*d=c^-1>;
// generators/relations

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