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G = C36⋊C12order 432 = 24·33

1st semidirect product of C36 and C12 acting via C12/C2=C6

metacyclic, supersoluble, monomial

Aliases: C361C12, C62.24D6, C4⋊(C9⋊C12), C4⋊Dic9⋊C3, (C2×C36).2C6, C18.4(C3×D4), C18.2(C3×Q8), C18.8(C2×C12), (C6×C12).10S3, C6.14(C3×D12), (C3×C6).21D12, (C3×C6).8Dic6, C6.8(C3×Dic6), C2.1(D36⋊C3), (C2×Dic9).2C6, C6.17(C6×Dic3), (C3×C12).3Dic3, C12.6(C3×Dic3), C32.(C4⋊Dic3), C2.2(C36.C6), 3- 1+22(C4⋊C4), (C4×3- 1+2)⋊1C4, (C2×3- 1+2).2Q8, (C2×3- 1+2).4D4, (C22×3- 1+2).4C22, C92(C3×C4⋊C4), C2.4(C2×C9⋊C12), (C2×C9⋊C12).2C2, (C2×C4).3(C9⋊C6), (C2×C18).4(C2×C6), (C2×C6).44(S3×C6), C22.5(C2×C9⋊C6), C3.3(C3×C4⋊Dic3), (C2×C12).13(C3×S3), (C3×C6).13(C2×Dic3), (C2×C4×3- 1+2).2C2, (C2×3- 1+2).8(C2×C4), SmallGroup(432,146)

Series: Derived Chief Lower central Upper central

C1C18 — C36⋊C12
C1C3C9C18C2×C18C22×3- 1+2C2×C9⋊C12 — C36⋊C12
C9C18 — C36⋊C12
C1C22C2×C4

Generators and relations for C36⋊C12
 G = < a,b | a36=b12=1, bab-1=a23 >

Subgroups: 262 in 86 conjugacy classes, 46 normal (32 characteristic)
C1, C2 [×3], C3, C3, C4 [×2], C4 [×2], C22, C6 [×3], C6 [×3], C2×C4, C2×C4 [×2], C9, C9, C32, Dic3 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6, C4⋊C4, C18 [×3], C18 [×3], C3×C6 [×3], C2×Dic3 [×2], C2×C12, C2×C12 [×3], 3- 1+2, Dic9 [×2], C36 [×2], C36 [×2], C2×C18, C2×C18, C3×Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3, C3×C4⋊C4, C2×3- 1+2 [×3], C2×Dic9 [×2], C2×C36, C2×C36, C6×Dic3 [×2], C6×C12, C9⋊C12 [×2], C4×3- 1+2 [×2], C22×3- 1+2, C4⋊Dic9, C3×C4⋊Dic3, C2×C9⋊C12 [×2], C2×C4×3- 1+2, C36⋊C12
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C9⋊C6, C3×Dic6, C3×D12, C6×Dic3, C9⋊C12 [×2], C2×C9⋊C6, C3×C4⋊Dic3, C36.C6, D36⋊C3, C2×C9⋊C12, C36⋊C12

Smallest permutation representation of C36⋊C12
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)
(1 138 69 100)(2 113 46 99 26 125 70 75 14 137 58 87)(3 124 59 98 15 112 71 86 27 136 47 74)(4 135 72 97)(5 110 49 96 29 122 37 108 17 134 61 84)(6 121 62 95 18 109 38 83 30 133 50 107)(7 132 39 94)(8 143 52 93 32 119 40 105 20 131 64 81)(9 118 65 92 21 142 41 80 33 130 53 104)(10 129 42 91)(11 140 55 90 35 116 43 102 23 128 67 78)(12 115 68 89 24 139 44 77 36 127 56 101)(13 126 45 88)(16 123 48 85)(19 120 51 82)(22 117 54 79)(25 114 57 76)(28 111 60 73)(31 144 63 106)(34 141 66 103)

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

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

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

62 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D···6I9A9B9C12A12B12C12D12E12F12G12H12I···12P18A···18I36A···36L
order12223334444446666···6999121212121212121212···1218···1836···36
size111123322181818182223···36662222666618···186···66···6

62 irreducible representations

dim111111112222222222222266666
type+++++--+-++-+-+
imageC1C2C2C3C4C6C6C12S3D4Q8Dic3D6C3×S3C3×D4C3×Q8Dic6D12C3×Dic3S3×C6C3×Dic6C3×D12C9⋊C6C9⋊C12C2×C9⋊C6C36.C6D36⋊C3
kernelC36⋊C12C2×C9⋊C12C2×C4×3- 1+2C4⋊Dic9C4×3- 1+2C2×Dic9C2×C36C36C6×C12C2×3- 1+2C2×3- 1+2C3×C12C62C2×C12C18C18C3×C6C3×C6C12C2×C6C6C6C2×C4C4C22C2C2
# reps121244281112122222424412122

Matrix representation of C36⋊C12 in GL8(𝔽37)

2334000000
029000000
000140000
006817000
008029000
0003008350
001225270298
003593014290
,
110000000
1526000000
0018002600
005193110016
00032513636
006001900
0032231924013
002449243213

G:=sub<GL(8,GF(37))| [23,0,0,0,0,0,0,0,34,29,0,0,0,0,0,0,0,0,0,6,8,0,12,35,0,0,14,8,0,30,25,9,0,0,0,17,29,0,27,30,0,0,0,0,0,8,0,14,0,0,0,0,0,35,29,29,0,0,0,0,0,0,8,0],[11,15,0,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,18,5,0,6,32,24,0,0,0,19,32,0,23,4,0,0,0,31,5,0,19,9,0,0,26,10,1,19,24,24,0,0,0,0,36,0,0,32,0,0,0,16,36,0,13,13] >;

C36⋊C12 in GAP, Magma, Sage, TeX

C_{36}\rtimes C_{12}
% in TeX

G:=Group("C36:C12");
// GroupNames label

G:=SmallGroup(432,146);
// by ID

G=gap.SmallGroup(432,146);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,10085,2035,292,14118]);
// Polycyclic

G:=Group<a,b|a^36=b^12=1,b*a*b^-1=a^23>;
// generators/relations

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