metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.7D4, C23.2Dic9, (C6×D4).1S3, (C2×D4).1D9, (C2×C4).3D18, (D4×C18).1C2, (C2×C12).45D6, C9⋊2(C4.D4), C4.Dic9⋊3C2, C4.12(C9⋊D4), C12.5(C3⋊D4), (C22×C18).2C4, C3.(C12.D4), (C2×C36).23C22, (C22×C6).6Dic3, C22.2(C2×Dic9), C18.13(C22⋊C4), C2.3(C18.D4), C6.14(C6.D4), (C2×C18).29(C2×C4), (C2×C6).33(C2×Dic3), SmallGroup(288,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.D4
G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=a-1, cac-1=a17, cbc-1=a27b3 >
Subgroups: 196 in 69 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, D4, C23, C9, C12, C2×C6, C2×C6, M4(2), C2×D4, C18, C18, C3⋊C8, C2×C12, C3×D4, C22×C6, C4.D4, C36, C2×C18, C2×C18, C4.Dic3, C6×D4, C9⋊C8, C2×C36, D4×C9, C22×C18, C12.D4, C4.Dic9, D4×C18, C36.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D9, C2×Dic3, C3⋊D4, C4.D4, Dic9, D18, C6.D4, C2×Dic9, C9⋊D4, C12.D4, C18.D4, C36.D4
►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 65 10 56 19 47 28 38)(2 64 11 55 20 46 29 37)(3 63 12 54 21 45 30 72)(4 62 13 53 22 44 31 71)(5 61 14 52 23 43 32 70)(6 60 15 51 24 42 33 69)(7 59 16 50 25 41 34 68)(8 58 17 49 26 40 35 67)(9 57 18 48 27 39 36 66)
(1 56 10 65 19 38 28 47)(2 37 11 46 20 55 29 64)(3 54 12 63 21 72 30 45)(4 71 13 44 22 53 31 62)(5 52 14 61 23 70 32 43)(6 69 15 42 24 51 33 60)(7 50 16 59 25 68 34 41)(8 67 17 40 26 49 35 58)(9 48 18 57 27 66 36 39)
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), (1,65,10,56,19,47,28,38)(2,64,11,55,20,46,29,37)(3,63,12,54,21,45,30,72)(4,62,13,53,22,44,31,71)(5,61,14,52,23,43,32,70)(6,60,15,51,24,42,33,69)(7,59,16,50,25,41,34,68)(8,58,17,49,26,40,35,67)(9,57,18,48,27,39,36,66), (1,56,10,65,19,38,28,47)(2,37,11,46,20,55,29,64)(3,54,12,63,21,72,30,45)(4,71,13,44,22,53,31,62)(5,52,14,61,23,70,32,43)(6,69,15,42,24,51,33,60)(7,50,16,59,25,68,34,41)(8,67,17,40,26,49,35,58)(9,48,18,57,27,66,36,39)>;
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), (1,65,10,56,19,47,28,38)(2,64,11,55,20,46,29,37)(3,63,12,54,21,45,30,72)(4,62,13,53,22,44,31,71)(5,61,14,52,23,43,32,70)(6,60,15,51,24,42,33,69)(7,59,16,50,25,41,34,68)(8,58,17,49,26,40,35,67)(9,57,18,48,27,39,36,66), (1,56,10,65,19,38,28,47)(2,37,11,46,20,55,29,64)(3,54,12,63,21,72,30,45)(4,71,13,44,22,53,31,62)(5,52,14,61,23,70,32,43)(6,69,15,42,24,51,33,60)(7,50,16,59,25,68,34,41)(8,67,17,40,26,49,35,58)(9,48,18,57,27,66,36,39) );
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)], [(1,65,10,56,19,47,28,38),(2,64,11,55,20,46,29,37),(3,63,12,54,21,45,30,72),(4,62,13,53,22,44,31,71),(5,61,14,52,23,43,32,70),(6,60,15,51,24,42,33,69),(7,59,16,50,25,41,34,68),(8,58,17,49,26,40,35,67),(9,57,18,48,27,39,36,66)], [(1,56,10,65,19,38,28,47),(2,37,11,46,20,55,29,64),(3,54,12,63,21,72,30,45),(4,71,13,44,22,53,31,62),(5,52,14,61,23,70,32,43),(6,69,15,42,24,51,33,60),(7,50,16,59,25,68,34,41),(8,67,17,40,26,49,35,58),(9,48,18,57,27,66,36,39)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 18A | ··· | 18I | 18J | ··· | 18U | 36A | ··· | 36F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 36 | 36 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | D9 | C3⋊D4 | D18 | Dic9 | C9⋊D4 | C4.D4 | C12.D4 | C36.D4 |
| kernel | C36.D4 | C4.Dic9 | D4×C18 | C22×C18 | C6×D4 | C36 | C2×C12 | C22×C6 | C2×D4 | C12 | C2×C4 | C23 | C4 | C9 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 3 | 4 | 3 | 6 | 12 | 1 | 2 | 6 |
Matrix representation of C36.D4 ►in GL4(𝔽73) generated by
| 55 | 34 | 0 | 0 |
| 11 | 18 | 0 | 0 |
| 27 | 18 | 0 | 4 |
| 57 | 18 | 69 | 0 |
| 1 | 0 | 0 | 63 |
| 0 | 0 | 1 | 72 |
| 44 | 0 | 0 | 72 |
| 0 | 1 | 0 | 72 |
| 1 | 0 | 63 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 1 | 72 | 0 |
| 44 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [55,11,27,57,34,18,18,18,0,0,0,69,0,0,4,0],[1,0,44,0,0,0,0,1,0,1,0,0,63,72,72,72],[1,0,0,44,0,0,1,0,63,72,72,72,0,1,0,0] >;
C36.D4 in GAP, Magma, Sage, TeX
C_{36}.D_4 % in TeX
G:=Group("C36.D4"); // GroupNames label
G:=SmallGroup(288,39);
// by ID
G=gap.SmallGroup(288,39);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,100,675,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^27*b^3>;
// generators/relations