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G = C4×C12⋊S3order 288 = 25·32

Direct product of C4 and C12⋊S3

direct product, metabelian, supersoluble, monomial

Aliases: C4×C12⋊S3, C125D12, C1228C2, C62.217C23, C129(C4×S3), (C4×C12)⋊9S3, C33(C4×D12), (C3×C12)⋊20D4, C3217(C4×D4), C425(C3⋊S3), C6.49(C2×D12), (C2×C12).356D6, C6.94(C4○D12), C12⋊Dic325C2, C6.11D1228C2, (C6×C12).286C22, C2.3(C12.59D6), C42(C4×C3⋊S3), C6.64(S3×C2×C4), (C3×C12)⋊17(C2×C4), C2.1(C2×C12⋊S3), (C3×C6).189(C2×D4), (C3×C6).95(C22×C4), (C2×C12⋊S3).17C2, (C3×C6).110(C4○D4), (C2×C6).234(C22×S3), C22.11(C22×C3⋊S3), (C22×C3⋊S3).80C22, (C2×C3⋊Dic3).152C22, C2.6(C2×C4×C3⋊S3), (C2×C4×C3⋊S3)⋊16C2, (C2×C3⋊S3)⋊9(C2×C4), (C2×C4).97(C2×C3⋊S3), SmallGroup(288,730)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C4×C12⋊S3
C1C3C32C3×C6C62C22×C3⋊S3C2×C12⋊S3 — C4×C12⋊S3
C32C3×C6 — C4×C12⋊S3
C1C2×C4C42

Generators and relations for C4×C12⋊S3
 G = < a,b,c,d | a4=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1124 in 282 conjugacy classes, 97 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×4], C4 [×3], C22, C22 [×8], S3 [×16], C6 [×12], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], C32, Dic3 [×8], C12 [×16], C12 [×4], D6 [×32], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×16], D12 [×16], C2×Dic3 [×8], C2×C12 [×12], C22×S3 [×8], C4×D4, C3⋊Dic3 [×2], C3×C12 [×4], C3×C12, C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C4⋊Dic3 [×4], D6⋊C4 [×8], C4×C12 [×4], S3×C2×C4 [×8], C2×D12 [×4], C4×C3⋊S3 [×4], C12⋊S3 [×4], C2×C3⋊Dic3 [×2], C6×C12 [×3], C22×C3⋊S3 [×2], C4×D12 [×4], C12⋊Dic3, C6.11D12 [×2], C122, C2×C4×C3⋊S3 [×2], C2×C12⋊S3, C4×C12⋊S3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], C23, D6 [×12], C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3 [×8], D12 [×8], C22×S3 [×4], C4×D4, C2×C3⋊S3 [×3], S3×C2×C4 [×4], C2×D12 [×4], C4○D12 [×4], C4×C3⋊S3 [×2], C12⋊S3 [×2], C22×C3⋊S3, C4×D12 [×4], C2×C4×C3⋊S3, C2×C12⋊S3, C12.59D6, C4×C12⋊S3

Smallest permutation representation of C4×C12⋊S3
On 144 points
Generators in S144
(1 112 32 103)(2 113 33 104)(3 114 34 105)(4 115 35 106)(5 116 36 107)(6 117 25 108)(7 118 26 97)(8 119 27 98)(9 120 28 99)(10 109 29 100)(11 110 30 101)(12 111 31 102)(13 50 90 137)(14 51 91 138)(15 52 92 139)(16 53 93 140)(17 54 94 141)(18 55 95 142)(19 56 96 143)(20 57 85 144)(21 58 86 133)(22 59 87 134)(23 60 88 135)(24 49 89 136)(37 82 124 72)(38 83 125 61)(39 84 126 62)(40 73 127 63)(41 74 128 64)(42 75 129 65)(43 76 130 66)(44 77 131 67)(45 78 132 68)(46 79 121 69)(47 80 122 70)(48 81 123 71)
(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 94 64)(2 95 65)(3 96 66)(4 85 67)(5 86 68)(6 87 69)(7 88 70)(8 89 71)(9 90 72)(10 91 61)(11 92 62)(12 93 63)(13 82 28)(14 83 29)(15 84 30)(16 73 31)(17 74 32)(18 75 33)(19 76 34)(20 77 35)(21 78 36)(22 79 25)(23 80 26)(24 81 27)(37 120 137)(38 109 138)(39 110 139)(40 111 140)(41 112 141)(42 113 142)(43 114 143)(44 115 144)(45 116 133)(46 117 134)(47 118 135)(48 119 136)(49 123 98)(50 124 99)(51 125 100)(52 126 101)(53 127 102)(54 128 103)(55 129 104)(56 130 105)(57 131 106)(58 132 107)(59 121 108)(60 122 97)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 84)(23 83)(24 82)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 136)(38 135)(39 134)(40 133)(41 144)(42 143)(43 142)(44 141)(45 140)(46 139)(47 138)(48 137)(49 124)(50 123)(51 122)(52 121)(53 132)(54 131)(55 130)(56 129)(57 128)(58 127)(59 126)(60 125)(61 88)(62 87)(63 86)(64 85)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(97 100)(98 99)(101 108)(102 107)(103 106)(104 105)(109 118)(110 117)(111 116)(112 115)(113 114)(119 120)

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B4C4D4E4F4G4H4I4J4K4L6A···6L12A···12AV
order1222222233334444444444446···612···12
size111118181818222211112222181818182···22···2

84 irreducible representations

dim11111112222222
type++++++++++
imageC1C2C2C2C2C2C4S3D4D6C4○D4C4×S3D12C4○D12
kernelC4×C12⋊S3C12⋊Dic3C6.11D12C122C2×C4×C3⋊S3C2×C12⋊S3C12⋊S3C4×C12C3×C12C2×C12C3×C6C12C12C6
# reps112121842122161616

Matrix representation of C4×C12⋊S3 in GL4(𝔽13) generated by

5000
0500
0050
0005
,
12100
12000
0098
0064
,
01200
11200
00110
00111
,
0100
1000
0098
0034
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[12,12,0,0,1,0,0,0,0,0,9,6,0,0,8,4],[0,1,0,0,12,12,0,0,0,0,1,1,0,0,10,11],[0,1,0,0,1,0,0,0,0,0,9,3,0,0,8,4] >;

C4×C12⋊S3 in GAP, Magma, Sage, TeX

C_4\times C_{12}\rtimes S_3
% in TeX

G:=Group("C4xC12:S3");
// GroupNames label

G:=SmallGroup(288,730);
// by ID

G=gap.SmallGroup(288,730);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^12=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^-1,d*c*d=c^-1>;
// generators/relations

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×
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