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G = C1226C2order 288 = 25·32

6th semidirect product of C122 and C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: C1226C2, C12.47D12, C62.219C23, (C4×C12)⋊7S3, C427(C3⋊S3), C6.51(C2×D12), (C3×C12).122D4, (C2×C12).357D6, C4.5(C12⋊S3), C32(C427S3), C6.95(C4○D12), C6.11D121C2, (C6×C12).287C22, C3212(C4.4D4), C2.7(C12.59D6), C2.6(C2×C12⋊S3), (C3×C6).191(C2×D4), (C2×C324Q8)⋊3C2, (C2×C12⋊S3).4C2, (C3×C6).111(C4○D4), (C2×C6).236(C22×S3), C22.37(C22×C3⋊S3), (C22×C3⋊S3).38C22, (C2×C3⋊Dic3).76C22, (C2×C4).65(C2×C3⋊S3), SmallGroup(288,732)

Series: Derived Chief Lower central Upper central

C1C62 — C1226C2
C1C3C32C3×C6C62C22×C3⋊S3C6.11D12 — C1226C2
C32C62 — C1226C2
C1C22C42

Generators and relations for C1226C2
 G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5b6, cbc=a6b5 >

Subgroups: 1036 in 228 conjugacy classes, 77 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×4], C4 [×2], C4 [×4], C22, C22 [×6], S3 [×8], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×8], C12 [×8], C12 [×8], D6 [×24], C2×C6 [×4], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×8], D12 [×8], C2×Dic3 [×8], C2×C12 [×12], C22×S3 [×8], C4.4D4, C3⋊Dic3 [×2], C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, D6⋊C4 [×16], C4×C12 [×4], C2×Dic6 [×4], C2×D12 [×4], C324Q8 [×2], C12⋊S3 [×2], C2×C3⋊Dic3 [×2], C6×C12, C6×C12 [×2], C22×C3⋊S3 [×2], C427S3 [×4], C6.11D12 [×4], C122, C2×C324Q8, C2×C12⋊S3, C1226C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C4○D4 [×2], C3⋊S3, D12 [×8], C22×S3 [×4], C4.4D4, C2×C3⋊S3 [×3], C2×D12 [×4], C4○D12 [×8], C12⋊S3 [×2], C22×C3⋊S3, C427S3 [×4], C2×C12⋊S3, C12.59D6 [×2], C1226C2

Smallest permutation representation of C1226C2
On 144 points
Generators in S144
(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 78 69 91 35 57 16 130 45 137 117 97)(2 79 70 92 36 58 17 131 46 138 118 98)(3 80 71 93 25 59 18 132 47 139 119 99)(4 81 72 94 26 60 19 121 48 140 120 100)(5 82 61 95 27 49 20 122 37 141 109 101)(6 83 62 96 28 50 21 123 38 142 110 102)(7 84 63 85 29 51 22 124 39 143 111 103)(8 73 64 86 30 52 23 125 40 144 112 104)(9 74 65 87 31 53 24 126 41 133 113 105)(10 75 66 88 32 54 13 127 42 134 114 106)(11 76 67 89 33 55 14 128 43 135 115 107)(12 77 68 90 34 56 15 129 44 136 116 108)
(2 21)(3 11)(4 19)(5 9)(6 17)(8 15)(10 13)(12 23)(14 18)(20 24)(25 43)(26 72)(27 41)(28 70)(29 39)(30 68)(31 37)(32 66)(33 47)(34 64)(35 45)(36 62)(38 118)(40 116)(42 114)(44 112)(46 110)(48 120)(49 80)(50 125)(51 78)(52 123)(53 76)(54 121)(55 74)(56 131)(57 84)(58 129)(59 82)(60 127)(61 113)(63 111)(65 109)(67 119)(69 117)(71 115)(73 102)(75 100)(77 98)(79 108)(81 106)(83 104)(85 91)(86 142)(87 89)(88 140)(90 138)(92 136)(93 95)(94 134)(96 144)(97 124)(99 122)(101 132)(103 130)(105 128)(107 126)(133 135)(137 143)(139 141)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A···4F4G4H6A···6L12A···12AV
order12222233334···4446···612···12
size1111363622222···236362···22···2

78 irreducible representations

dim11111222222
type+++++++++
imageC1C2C2C2C2S3D4D6C4○D4D12C4○D12
kernelC1226C2C6.11D12C122C2×C324Q8C2×C12⋊S3C4×C12C3×C12C2×C12C3×C6C12C6
# reps14111421241632

Matrix representation of C1226C2 in GL4(𝔽13) generated by

0500
8800
0092
001111
,
11900
4200
0058
0050
,
1000
121200
0001
0010
G:=sub<GL(4,GF(13))| [0,8,0,0,5,8,0,0,0,0,9,11,0,0,2,11],[11,4,0,0,9,2,0,0,0,0,5,5,0,0,8,0],[1,12,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;

C1226C2 in GAP, Magma, Sage, TeX

C_{12}^2\rtimes_6C_2
% in TeX

G:=Group("C12^2:6C2");
// GroupNames label

G:=SmallGroup(288,732);
// by ID

G=gap.SmallGroup(288,732);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,100,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^5*b^6,c*b*c=a^6*b^5>;
// generators/relations

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