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G = C123D12order 288 = 25·32

3rd semidirect product of C12 and D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C123D12, C62.239C23, (C3×C12)⋊12D4, (C2×C12).34D6, C6.116(S3×D4), C6.54(C2×D12), C42(C12⋊S3), C32(C12⋊D4), C6.11D127C2, C3219(C4⋊D4), (C6×C12).16C22, C6.51(Q83S3), C2.6(C12.26D6), (C3×C4⋊C4)⋊6S3, C4⋊C43(C3⋊S3), (C2×C3⋊S3)⋊12D4, C2.13(D4×C3⋊S3), (C2×C12⋊S3)⋊6C2, (C32×C4⋊C4)⋊15C2, C2.9(C2×C12⋊S3), (C3×C6).194(C2×D4), (C3×C6).161(C4○D4), (C2×C6).256(C22×S3), C22.50(C22×C3⋊S3), (C22×C3⋊S3).87C22, (C2×C3⋊Dic3).160C22, (C2×C4×C3⋊S3)⋊2C2, (C2×C4).12(C2×C3⋊S3), SmallGroup(288,752)

Series: Derived Chief Lower central Upper central

C1C62 — C123D12
C1C3C32C3×C6C62C22×C3⋊S3C2×C4×C3⋊S3 — C123D12
C32C62 — C123D12
C1C22C4⋊C4

Generators and relations for C123D12
 G = < a,b,c | a12=b12=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >

Subgroups: 1388 in 282 conjugacy classes, 79 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×3], C22, C22 [×10], S3 [×16], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], C32, Dic3 [×4], C12 [×8], C12 [×8], D6 [×40], C2×C6 [×4], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×8], D12 [×24], C2×Dic3 [×4], C2×C12 [×12], C22×S3 [×12], C4⋊D4, C3⋊Dic3, C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×8], C62, D6⋊C4 [×8], C3×C4⋊C4 [×4], S3×C2×C4 [×4], C2×D12 [×12], C4×C3⋊S3 [×2], C12⋊S3 [×6], C2×C3⋊Dic3, C6×C12, C6×C12 [×2], C22×C3⋊S3, C22×C3⋊S3 [×2], C12⋊D4 [×4], C6.11D12 [×2], C32×C4⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C2×C12⋊S3 [×2], C123D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×4], C23, D6 [×12], C2×D4 [×2], C4○D4, C3⋊S3, D12 [×8], C22×S3 [×4], C4⋊D4, C2×C3⋊S3 [×3], C2×D12 [×4], S3×D4 [×4], Q83S3 [×4], C12⋊S3 [×2], C22×C3⋊S3, C12⋊D4 [×4], C2×C12⋊S3, D4×C3⋊S3, C12.26D6, C123D12

Smallest permutation representation of C123D12
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 48 33 67 91 56 119 21 80 106 138 132)(2 43 34 62 92 51 120 16 81 101 139 127)(3 38 35 69 93 58 109 23 82 108 140 122)(4 45 36 64 94 53 110 18 83 103 141 129)(5 40 25 71 95 60 111 13 84 98 142 124)(6 47 26 66 96 55 112 20 73 105 143 131)(7 42 27 61 85 50 113 15 74 100 144 126)(8 37 28 68 86 57 114 22 75 107 133 121)(9 44 29 63 87 52 115 17 76 102 134 128)(10 39 30 70 88 59 116 24 77 97 135 123)(11 46 31 65 89 54 117 19 78 104 136 130)(12 41 32 72 90 49 118 14 79 99 137 125)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 17)(14 16)(18 24)(19 23)(20 22)(37 47)(38 46)(39 45)(40 44)(41 43)(49 101)(50 100)(51 99)(52 98)(53 97)(54 108)(55 107)(56 106)(57 105)(58 104)(59 103)(60 102)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)(73 114)(74 113)(75 112)(76 111)(77 110)(78 109)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 144)(86 143)(87 142)(88 141)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B4C4D4E4F6A···6L12A···12X
order1222222233334444446···612···12
size1111181836362222224418182···24···4

54 irreducible representations

dim1111122222244
type++++++++++++
imageC1C2C2C2C2S3D4D4D6C4○D4D12S3×D4Q83S3
kernelC123D12C6.11D12C32×C4⋊C4C2×C4×C3⋊S3C2×C12⋊S3C3×C4⋊C4C3×C12C2×C3⋊S3C2×C12C3×C6C12C6C6
# reps121134221221644

Matrix representation of C123D12 in GL8(𝔽13)

10000000
01000000
000120000
00110000
000012000
000001200
000000117
00000032
,
1210000000
51000000
00110000
001200000
00000100
000012100
000000120
00000051
,
120000000
51000000
0012120000
00010000
000001200
000012000
00000010
000000812

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,11,3,0,0,0,0,0,0,7,2],[12,5,0,0,0,0,0,0,10,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1],[12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12] >;

C123D12 in GAP, Magma, Sage, TeX

C_{12}\rtimes_3D_{12}
% in TeX

G:=Group("C12:3D12");
// GroupNames label

G:=SmallGroup(288,752);
// by ID

G=gap.SmallGroup(288,752);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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