metacyclic, supersoluble, monomial
Aliases: C36⋊1C12, 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+2⋊2(C4⋊C4), (C4×3- 1+2)⋊1C4, (C2×3- 1+2).2Q8, (C2×3- 1+2).4D4, (C22×3- 1+2).4C22, C9⋊2(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
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, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, 3- 1+2, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C2×3- 1+2, C2×Dic9, C2×C36, C2×C36, C6×Dic3, C6×C12, C9⋊C12, C4×3- 1+2, C22×3- 1+2, C4⋊Dic9, C3×C4⋊Dic3, C2×C9⋊C12, C2×C4×3- 1+2, C36⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3, S3×C6, C4⋊Dic3, C3×C4⋊C4, C9⋊C6, C3×Dic6, C3×D12, C6×Dic3, C9⋊C12, C2×C9⋊C6, C3×C4⋊Dic3, C36.C6, D36⋊C3, C2×C9⋊C12, C36⋊C12
►On 144 points
(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)
(1 105 47 136)(2 80 60 135 26 92 48 111 14 104 72 123)(3 91 37 134 15 79 49 122 27 103 61 110)(4 102 50 133)(5 77 63 132 29 89 51 144 17 101 39 120)(6 88 40 131 18 76 52 119 30 100 64 143)(7 99 53 130)(8 74 66 129 32 86 54 141 20 98 42 117)(9 85 43 128 21 73 55 116 33 97 67 140)(10 96 56 127)(11 107 69 126 35 83 57 138 23 95 45 114)(12 82 46 125 24 106 58 113 36 94 70 137)(13 93 59 124)(16 90 62 121)(19 87 65 118)(22 84 68 115)(25 81 71 112)(28 78 38 109)(31 75 41 142)(34 108 44 139)
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), (1,105,47,136)(2,80,60,135,26,92,48,111,14,104,72,123)(3,91,37,134,15,79,49,122,27,103,61,110)(4,102,50,133)(5,77,63,132,29,89,51,144,17,101,39,120)(6,88,40,131,18,76,52,119,30,100,64,143)(7,99,53,130)(8,74,66,129,32,86,54,141,20,98,42,117)(9,85,43,128,21,73,55,116,33,97,67,140)(10,96,56,127)(11,107,69,126,35,83,57,138,23,95,45,114)(12,82,46,125,24,106,58,113,36,94,70,137)(13,93,59,124)(16,90,62,121)(19,87,65,118)(22,84,68,115)(25,81,71,112)(28,78,38,109)(31,75,41,142)(34,108,44,139)>;
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), (1,105,47,136)(2,80,60,135,26,92,48,111,14,104,72,123)(3,91,37,134,15,79,49,122,27,103,61,110)(4,102,50,133)(5,77,63,132,29,89,51,144,17,101,39,120)(6,88,40,131,18,76,52,119,30,100,64,143)(7,99,53,130)(8,74,66,129,32,86,54,141,20,98,42,117)(9,85,43,128,21,73,55,116,33,97,67,140)(10,96,56,127)(11,107,69,126,35,83,57,138,23,95,45,114)(12,82,46,125,24,106,58,113,36,94,70,137)(13,93,59,124)(16,90,62,121)(19,87,65,118)(22,84,68,115)(25,81,71,112)(28,78,38,109)(31,75,41,142)(34,108,44,139) );
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)], [(1,105,47,136),(2,80,60,135,26,92,48,111,14,104,72,123),(3,91,37,134,15,79,49,122,27,103,61,110),(4,102,50,133),(5,77,63,132,29,89,51,144,17,101,39,120),(6,88,40,131,18,76,52,119,30,100,64,143),(7,99,53,130),(8,74,66,129,32,86,54,141,20,98,42,117),(9,85,43,128,21,73,55,116,33,97,67,140),(10,96,56,127),(11,107,69,126,35,83,57,138,23,95,45,114),(12,82,46,125,24,106,58,113,36,94,70,137),(13,93,59,124),(16,90,62,121),(19,87,65,118),(22,84,68,115),(25,81,71,112),(28,78,38,109),(31,75,41,142),(34,108,44,139)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | ··· | 6I | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12P | 18A | ··· | 18I | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | ··· | 18 | 6 | ··· | 6 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | - | + | - | + | + | - | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | C3×D4 | C3×Q8 | Dic6 | D12 | C3×Dic3 | S3×C6 | C3×Dic6 | C3×D12 | C9⋊C6 | C9⋊C12 | C2×C9⋊C6 | C36.C6 | D36⋊C3 |
| kernel | C36⋊C12 | C2×C9⋊C12 | C2×C4×3- 1+2 | C4⋊Dic9 | C4×3- 1+2 | C2×Dic9 | C2×C36 | C36 | C6×C12 | C2×3- 1+2 | C2×3- 1+2 | C3×C12 | C62 | C2×C12 | C18 | C18 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C36⋊C12 ►in GL8(𝔽37)
| 23 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 29 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 8 | 17 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 29 | 0 | 0 | 0 |
| 0 | 0 | 0 | 30 | 0 | 8 | 35 | 0 |
| 0 | 0 | 12 | 25 | 27 | 0 | 29 | 8 |
| 0 | 0 | 35 | 9 | 30 | 14 | 29 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 5 | 19 | 31 | 10 | 0 | 16 |
| 0 | 0 | 0 | 32 | 5 | 1 | 36 | 36 |
| 0 | 0 | 6 | 0 | 0 | 19 | 0 | 0 |
| 0 | 0 | 32 | 23 | 19 | 24 | 0 | 13 |
| 0 | 0 | 24 | 4 | 9 | 24 | 32 | 13 |
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