metacyclic, supersoluble, monomial
Aliases: C36.1C12, C62.4Dic3, 3- 1+2⋊2M4(2), C9⋊C8⋊5C6, C9⋊C24⋊5C2, C4.(C9⋊C12), C4.Dic9⋊C3, (C6×C12).6S3, (C2×C36).1C6, C12.95(S3×C6), C18.6(C2×C12), C36.16(C2×C6), (C2×C18).2C12, C22.(C9⋊C12), (C3×C12).63D6, C9⋊2(C3×M4(2)), C6.15(C6×Dic3), C12.5(C3×Dic3), (C3×C12).2Dic3, C32.(C4.Dic3), (C4×3- 1+2).1C4, (C22×3- 1+2).2C4, (C4×3- 1+2).15C22, C4.15(C2×C9⋊C6), C2.3(C2×C9⋊C12), (C2×C4).2(C9⋊C6), (C2×C12).24(C3×S3), C3.3(C3×C4.Dic3), (C2×C6).17(C3×Dic3), (C3×C6).11(C2×Dic3), (C2×C4×3- 1+2).1C2, (C2×3- 1+2).6(C2×C4), SmallGroup(432,143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.C12
G = < a,b | a36=1, b12=a18, bab-1=a23 >
Subgroups: 158 in 68 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, 3- 1+2, C36, C36, C2×C18, C2×C18, C3×C12, C62, C4.Dic3, C3×M4(2), C2×3- 1+2, C2×3- 1+2, C9⋊C8, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C4×3- 1+2, C22×3- 1+2, C4.Dic9, C3×C4.Dic3, C9⋊C24, C2×C4×3- 1+2, C36.C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, M4(2), C3×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), C9⋊C6, C6×Dic3, C9⋊C12, C2×C9⋊C6, C3×C4.Dic3, C2×C9⋊C12, C36.C12
►On 72 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)
(1 47 10 38 19 65 28 56)(2 58 23 37 8 52 29 67 14 46 35 61 20 40 5 55 26 70 11 49 32 64 17 43)(3 69 36 72 33 39 30 42 27 45 24 48 21 51 18 54 15 57 12 60 9 63 6 66)(4 44 13 71 22 62 31 53)(7 41 16 68 25 59 34 50)
G:=sub<Sym(72)| (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), (1,47,10,38,19,65,28,56)(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43)(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66)(4,44,13,71,22,62,31,53)(7,41,16,68,25,59,34,50)>;
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), (1,47,10,38,19,65,28,56)(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43)(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66)(4,44,13,71,22,62,31,53)(7,41,16,68,25,59,34,50) );
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)], [(1,47,10,38,19,65,28,56),(2,58,23,37,8,52,29,67,14,46,35,61,20,40,5,55,26,70,11,49,32,64,17,43),(3,69,36,72,33,39,30,42,27,45,24,48,21,51,18,54,15,57,12,60,9,63,6,66),(4,44,13,71,22,62,31,53),(7,41,16,68,25,59,34,50)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 18A | ··· | 18I | 24A | ··· | 24H | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 18 | ··· | 18 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | Dic3 | M4(2) | C3×S3 | C3×Dic3 | S3×C6 | C3×Dic3 | C3×M4(2) | C4.Dic3 | C3×C4.Dic3 | C9⋊C6 | C9⋊C12 | C2×C9⋊C6 | C9⋊C12 | C36.C12 |
| kernel | C36.C12 | C9⋊C24 | C2×C4×3- 1+2 | C4.Dic9 | C4×3- 1+2 | C22×3- 1+2 | C9⋊C8 | C2×C36 | C36 | C2×C18 | C6×C12 | C3×C12 | C3×C12 | C62 | 3- 1+2 | C2×C12 | C12 | C12 | C2×C6 | C9 | C32 | C3 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 4 |
Matrix representation of C36.C12 ►in GL6(𝔽73)
| 70 | 30 | 0 | 0 | 0 | 0 |
| 0 | 3 | 70 | 0 | 0 | 0 |
| 24 | 3 | 0 | 0 | 0 | 0 |
| 21 | 9 | 0 | 0 | 0 | 3 |
| 22 | 0 | 0 | 24 | 0 | 0 |
| 21 | 0 | 9 | 0 | 24 | 0 |
| 20 | 0 | 0 | 71 | 0 | 0 |
| 0 | 0 | 20 | 29 | 60 | 0 |
| 59 | 14 | 0 | 29 | 0 | 42 |
| 4 | 0 | 0 | 53 | 0 | 0 |
| 19 | 29 | 0 | 0 | 0 | 59 |
| 37 | 0 | 68 | 0 | 53 | 0 |
G:=sub<GL(6,GF(73))| [70,0,24,21,22,21,30,3,3,9,0,0,0,70,0,0,0,9,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,3,0,0],[20,0,59,4,19,37,0,0,14,0,29,0,0,20,0,0,0,68,71,29,29,53,0,0,0,60,0,0,0,53,0,0,42,0,59,0] >;
C36.C12 in GAP, Magma, Sage, TeX
C_{36}.C_{12} % in TeX
G:=Group("C36.C12"); // GroupNames label
G:=SmallGroup(432,143);
// by ID
G=gap.SmallGroup(432,143);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,10085,2035,292,14118]);
// Polycyclic
G:=Group<a,b|a^36=1,b^12=a^18,b*a*b^-1=a^23>;
// generators/relations