metacyclic, supersoluble, monomial
Aliases: C36.1C6, Dic18⋊C3, Dic9.C6, 3- 1+2⋊Q8, C32.Dic6, C9⋊(C3×Q8), C9⋊C12.C2, C4.(C9⋊C6), (C3×C6).9D6, C12.5(C3×S3), C6.12(S3×C6), C18.1(C2×C6), (C3×C12).2S3, C3.3(C3×Dic6), (C4×3- 1+2).1C2, (C2×3- 1+2).1C22, C2.3(C2×C9⋊C6), SmallGroup(216,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.C6
G = < a,b | a36=1, b6=a18, bab-1=a11 >
Smallest permutation representation of C36.C6
►On 72 points
Generators in S72
(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 42 19 60)(2 65 8 59 14 53 20 47 26 41 32 71)(3 52 33 58 27 64 21 70 15 40 9 46)(4 39 22 57)(5 62 11 56 17 50 23 44 29 38 35 68)(6 49 36 55 30 61 24 67 18 37 12 43)(7 72 25 54)(10 69 28 51)(13 66 31 48)(16 63 34 45)
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), (1,42,19,60)(2,65,8,59,14,53,20,47,26,41,32,71)(3,52,33,58,27,64,21,70,15,40,9,46)(4,39,22,57)(5,62,11,56,17,50,23,44,29,38,35,68)(6,49,36,55,30,61,24,67,18,37,12,43)(7,72,25,54)(10,69,28,51)(13,66,31,48)(16,63,34,45)>;
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), (1,42,19,60)(2,65,8,59,14,53,20,47,26,41,32,71)(3,52,33,58,27,64,21,70,15,40,9,46)(4,39,22,57)(5,62,11,56,17,50,23,44,29,38,35,68)(6,49,36,55,30,61,24,67,18,37,12,43)(7,72,25,54)(10,69,28,51)(13,66,31,48)(16,63,34,45) );
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)], [(1,42,19,60),(2,65,8,59,14,53,20,47,26,41,32,71),(3,52,33,58,27,64,21,70,15,40,9,46),(4,39,22,57),(5,62,11,56,17,50,23,44,29,38,35,68),(6,49,36,55,30,61,24,67,18,37,12,43),(7,72,25,54),(10,69,28,51),(13,66,31,48),(16,63,34,45)]])
C36.C6 is a maximal subgroup of
C72.C6 C72⋊2C6 Dic18⋊C6 Dic18.C6 D36⋊6C6 Dic18⋊2C6 Q8×C9⋊C6
C36.C6 is a maximal quotient of Dic9⋊C12 C36⋊C12
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | 18B | 18C | 36A | ··· | 36F |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 18 | 18 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 2 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | 6 | 6 | 6 | ··· | 6 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | Q8 | D6 | C3×S3 | C3×Q8 | Dic6 | S3×C6 | C3×Dic6 | C9⋊C6 | C2×C9⋊C6 | C36.C6 |
| kernel | C36.C6 | C9⋊C12 | C4×3- 1+2 | Dic18 | Dic9 | C36 | C3×C12 | 3- 1+2 | C3×C6 | C12 | C9 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C36.C6 ►in GL8(𝔽37)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 35 | 36 | 0 | 0 |
| 0 | 0 | 1 | 1 | 36 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 36 | 1 | 1 |
| 0 | 0 | 1 | 1 | 36 | 36 | 36 | 0 |
| 0 | 0 | 0 | 1 | 36 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 | 36 | 36 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 25 | 33 |
| 0 | 0 | 29 | 25 | 0 | 0 | 8 | 12 |
| 0 | 0 | 29 | 25 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 33 | 4 | 29 | 0 | 0 |
G:=sub<GL(8,GF(37))| [0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0,35,36,36,36,36,36,0,0,36,35,36,36,36,36,0,0,0,0,1,36,0,0,0,0,0,0,1,0,0,0],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,4,29,4,29,29,0,0,0,25,33,0,25,25,33,0,0,0,0,0,0,8,4,0,0,0,0,0,0,12,29,0,0,0,0,25,8,0,0,0,0,0,0,33,12,0,0] >;
C36.C6 in GAP, Magma, Sage, TeX
C_{36}.C_6 % in TeX
G:=Group("C36.C6"); // GroupNames label
G:=SmallGroup(216,52);
// by ID
G=gap.SmallGroup(216,52);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,3604,736,208,5189]);
// Polycyclic
G:=Group<a,b|a^36=1,b^6=a^18,b*a*b^-1=a^11>;
// generators/relations
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