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G = C12.20D12order 288 = 25·32

20th non-split extension by C12 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.20D12, (C2×C12).91D6, C6.28(D6⋊C4), (C3×C12).152D4, C62.41(C2×C4), (C6×C12).58C22, C4.12(C12⋊S3), C32(C12.47D4), C12.117(C3⋊D4), C12.58D6.5C2, M4(2).2(C3⋊S3), C326(C4.10D4), (C3×M4(2)).12S3, C4.22(C327D4), (C32×M4(2)).4C2, C2.10(C6.11D12), (C2×C6).15(C4×S3), C22.5(C4×C3⋊S3), (C2×C3⋊Dic3).2C4, (C3×C6).59(C22⋊C4), (C2×C324Q8).10C2, (C2×C4).2(C2×C3⋊S3), SmallGroup(288,299)

Series: Derived Chief Lower central Upper central

C1C62 — C12.20D12
C1C3C32C3×C6C3×C12C6×C12C2×C324Q8 — C12.20D12
C32C3×C6C62 — C12.20D12
C1C2C2×C4M4(2)

Generators and relations for C12.20D12
 G = < a,b,c | a12=1, b12=a6, c2=a3, bab-1=a7, cac-1=a5, cbc-1=a3b11 >

Subgroups: 396 in 114 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2, C3 [×4], C4 [×2], C4 [×2], C22, C6 [×4], C6 [×4], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, Dic3 [×8], C12 [×8], C2×C6 [×4], M4(2), M4(2), C2×Q8, C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], Dic6 [×8], C2×Dic3 [×8], C2×C12 [×4], C4.10D4, C3⋊Dic3 [×2], C3×C12 [×2], C62, C4.Dic3 [×4], C3×M4(2) [×4], C2×Dic6 [×4], C324C8, C3×C24, C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C12.47D4 [×4], C12.58D6, C32×M4(2), C2×C324Q8, C12.20D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C4.10D4, C2×C3⋊S3, D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C12.47D4 [×4], C6.11D12, C12.20D12

Smallest permutation representation of C12.20D12
On 144 points
Generators in S144
(1 107 42 7 113 48 13 119 30 19 101 36)(2 120 43 20 114 37 14 108 31 8 102 25)(3 109 44 9 115 26 15 97 32 21 103 38)(4 98 45 22 116 39 16 110 33 10 104 27)(5 111 46 11 117 28 17 99 34 23 105 40)(6 100 47 24 118 41 18 112 35 12 106 29)(49 132 76 67 126 94 61 144 88 55 138 82)(50 121 77 56 127 83 62 133 89 68 139 95)(51 134 78 69 128 96 63 122 90 57 140 84)(52 123 79 58 129 85 64 135 91 70 141 73)(53 136 80 71 130 74 65 124 92 59 142 86)(54 125 81 60 131 87 66 137 93 72 143 75)
(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 80 7 74 13 92 19 86)(2 73 20 79 14 85 8 91)(3 78 9 96 15 90 21 84)(4 95 22 77 16 83 10 89)(5 76 11 94 17 88 23 82)(6 93 24 75 18 81 12 87)(25 64 43 70 37 52 31 58)(26 69 32 63 38 57 44 51)(27 62 45 68 39 50 33 56)(28 67 34 61 40 55 46 49)(29 60 47 66 41 72 35 54)(30 65 36 59 42 53 48 71)(97 134 103 128 109 122 115 140)(98 127 116 133 110 139 104 121)(99 132 105 126 111 144 117 138)(100 125 118 131 112 137 106 143)(101 130 107 124 113 142 119 136)(102 123 120 129 114 135 108 141)

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

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

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

51 conjugacy classes

class 1 2A2B3A3B3C3D4A4B4C4D6A6B6C6D6E6F6G6H8A8B8C8D12A···12H12I12J12K12L24A···24P
order1223333444466666666888812···121212121224···24
size1122222223636222244444436362···244444···4

51 irreducible representations

dim1111122222244
type++++++++--
imageC1C2C2C2C4S3D4D6D12C3⋊D4C4×S3C4.10D4C12.47D4
kernelC12.20D12C12.58D6C32×M4(2)C2×C324Q8C2×C3⋊Dic3C3×M4(2)C3×C12C2×C12C12C12C2×C6C32C3
# reps1111442488818

Matrix representation of C12.20D12 in GL6(𝔽73)

36260000
47360000
0066700
00665900
006060766
00130714
,
010000
7200000
002727721
004607271
0042174646
005625270
,
2560000
56710000
0011134049
00262933
0027171113
006346262

G:=sub<GL(6,GF(73))| [36,47,0,0,0,0,26,36,0,0,0,0,0,0,66,66,60,13,0,0,7,59,60,0,0,0,0,0,7,7,0,0,0,0,66,14],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,27,46,42,56,0,0,27,0,17,25,0,0,72,72,46,27,0,0,1,71,46,0],[2,56,0,0,0,0,56,71,0,0,0,0,0,0,11,2,27,63,0,0,13,62,17,46,0,0,40,9,11,2,0,0,49,33,13,62] >;

C12.20D12 in GAP, Magma, Sage, TeX

C_{12}._{20}D_{12}
% in TeX

G:=Group("C12.20D12");
// GroupNames label

G:=SmallGroup(288,299);
// by ID

G=gap.SmallGroup(288,299);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^3,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations

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