metabelian, supersoluble, monomial
Aliases: C32⋊C36, C33.1C12, (C3×C6).C18, C3⋊Dic3⋊C9, C6.2(S3×C9), C32⋊C9⋊1C4, (C3×C18).1S3, (C3×C9)⋊1Dic3, C2.(C32⋊C18), (C32×C6).1C6, C3.2(C9×Dic3), C6.10(C32⋊C6), C3.5(C32⋊C12), C32.12(C3×Dic3), (C3×C3⋊Dic3).C3, (C3×C6).26(C3×S3), (C2×C32⋊C9).1C2, SmallGroup(324,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32⋊C36 |
Generators and relations for C32⋊C36
G = < a,b,c | a3=b3=c36=1, ab=ba, cac-1=a-1b, cbc-1=b-1 >
Smallest permutation representation of C32⋊C36
►On 36 points
Generators in S36
(2 26 14)(3 27 15)(5 17 29)(6 18 30)(8 32 20)(9 33 21)(11 23 35)(12 24 36)
(1 25 13)(2 14 26)(3 27 15)(4 16 28)(5 29 17)(6 18 30)(7 31 19)(8 20 32)(9 33 21)(10 22 34)(11 35 23)(12 24 36)
(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)
G:=sub<Sym(36)| (2,26,14)(3,27,15)(5,17,29)(6,18,30)(8,32,20)(9,33,21)(11,23,35)(12,24,36), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36), (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)>;
G:=Group( (2,26,14)(3,27,15)(5,17,29)(6,18,30)(8,32,20)(9,33,21)(11,23,35)(12,24,36), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36), (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) );
G=PermutationGroup([[(2,26,14),(3,27,15),(5,17,29),(6,18,30),(8,32,20),(9,33,21),(11,23,35),(12,24,36)], [(1,25,13),(2,14,26),(3,27,15),(4,16,28),(5,29,17),(6,18,30),(7,31,19),(8,20,32),(9,33,21),(10,22,34),(11,35,23),(12,24,36)], [(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)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 18G | ··· | 18L | 36A | ··· | 36L |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | 9 | 9 | 9 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C36 | S3 | Dic3 | C3×S3 | C3×Dic3 | S3×C9 | C9×Dic3 | C32⋊C6 | C32⋊C12 | C32⋊C18 | C32⋊C36 |
| kernel | C32⋊C36 | C2×C32⋊C9 | C3×C3⋊Dic3 | C32⋊C9 | C32×C6 | C3⋊Dic3 | C33 | C3×C6 | C32 | C3×C18 | C3×C9 | C3×C6 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 1 | 2 | 2 |
Matrix representation of C32⋊C36 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 26 |
| 26 | 0 | 0 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 0 | 26 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,26,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,26],[26,0,0,0,0,0,0,26,0,0,0,0,0,0,26,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[0,0,0,0,0,36,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0,0,26,0,0,0,0,0,0,26,0,0,0,0] >;
C32⋊C36 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_{36} % in TeX
G:=Group("C3^2:C36"); // GroupNames label
G:=SmallGroup(324,7);
// by ID
G=gap.SmallGroup(324,7);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,2164,2170,7781]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^36=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=b^-1>;
// generators/relations
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