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G = C1222C2order 288 = 25·32

2nd semidirect product of C122 and C2 acting faithfully

metabelian, supersoluble, monomial

Aliases: C1222C2, C62.220C23, (C4×C12)⋊5S3, C423(C3⋊S3), (C2×C12).358D6, C33(C423S3), C6.96(C4○D12), C6.Dic61C2, (C6×C12).288C22, C329(C422C2), C6.11D12.1C2, C2.8(C12.59D6), (C3×C6).112(C4○D4), (C2×C6).237(C22×S3), C22.38(C22×C3⋊S3), (C22×C3⋊S3).39C22, (C2×C3⋊Dic3).77C22, (C2×C4).66(C2×C3⋊S3), SmallGroup(288,733)

Series: Derived Chief Lower central Upper central

C1C62 — C1222C2
C1C3C32C3×C6C62C22×C3⋊S3C6.11D12 — C1222C2
C32C62 — C1222C2
C1C22C42

Generators and relations for C1222C2
 G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a5b6, cbc=a6b-1 >

Subgroups: 684 in 180 conjugacy classes, 65 normal (8 characteristic)
C1, C2 [×3], C2, C3 [×4], C4 [×6], C22, C22 [×3], S3 [×4], C6 [×12], C2×C4 [×3], C2×C4 [×3], C23, C32, Dic3 [×12], C12 [×12], D6 [×12], C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4 [×3], C3⋊S3, C3×C6 [×3], C2×Dic3 [×12], C2×C12 [×12], C22×S3 [×4], C422C2, C3⋊Dic3 [×3], C3×C12 [×3], C2×C3⋊S3 [×3], C62, Dic3⋊C4 [×12], D6⋊C4 [×12], C4×C12 [×4], C2×C3⋊Dic3 [×3], C6×C12 [×3], C22×C3⋊S3, C423S3 [×4], C6.Dic6 [×3], C6.11D12 [×3], C122, C1222C2
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C4○D4 [×3], C3⋊S3, C22×S3 [×4], C422C2, C2×C3⋊S3 [×3], C4○D12 [×12], C22×C3⋊S3, C423S3 [×4], C12.59D6 [×3], C1222C2

Smallest permutation representation of C1222C2
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 14 79 42 105 62 91 121 26 142 59 113)(2 15 80 43 106 63 92 122 27 143 60 114)(3 16 81 44 107 64 93 123 28 144 49 115)(4 17 82 45 108 65 94 124 29 133 50 116)(5 18 83 46 97 66 95 125 30 134 51 117)(6 19 84 47 98 67 96 126 31 135 52 118)(7 20 73 48 99 68 85 127 32 136 53 119)(8 21 74 37 100 69 86 128 33 137 54 120)(9 22 75 38 101 70 87 129 34 138 55 109)(10 23 76 39 102 71 88 130 35 139 56 110)(11 24 77 40 103 72 89 131 36 140 57 111)(12 13 78 41 104 61 90 132 25 141 58 112)
(2 96)(3 11)(4 94)(5 9)(6 92)(8 90)(10 88)(12 86)(13 63)(14 119)(15 61)(16 117)(17 71)(18 115)(19 69)(20 113)(21 67)(22 111)(23 65)(24 109)(25 54)(26 105)(27 52)(28 103)(29 50)(30 101)(31 60)(32 99)(33 58)(34 97)(35 56)(36 107)(37 47)(38 140)(39 45)(40 138)(41 43)(42 136)(44 134)(46 144)(48 142)(49 77)(51 75)(53 73)(55 83)(57 81)(59 79)(62 127)(64 125)(66 123)(68 121)(70 131)(72 129)(74 104)(76 102)(78 100)(80 98)(82 108)(84 106)(87 95)(89 93)(110 124)(112 122)(114 132)(116 130)(118 128)(120 126)(133 139)(135 137)(141 143)

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

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

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

78 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A···4F4G4H4I6A···6L12A···12AV
order1222233334···44446···612···12
size11113622222···23636362···22···2

78 irreducible representations

dim11112222
type++++++
imageC1C2C2C2S3D6C4○D4C4○D12
kernelC1222C2C6.Dic6C6.11D12C122C4×C12C2×C12C3×C6C6
# reps1331412648

Matrix representation of C1222C2 in GL4(𝔽13) generated by

101000
3700
00114
0092
,
5800
5000
0033
00106
,
0100
1000
00121
0001
G:=sub<GL(4,GF(13))| [10,3,0,0,10,7,0,0,0,0,11,9,0,0,4,2],[5,5,0,0,8,0,0,0,0,0,3,10,0,0,3,6],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,1,1] >;

C1222C2 in GAP, Magma, Sage, TeX

C_{12}^2\rtimes_2C_2
% in TeX

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

G:=SmallGroup(288,733);
// by ID

G=gap.SmallGroup(288,733);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,590,100,2693,9414]);
// 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^-1>;
// generators/relations

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