non-abelian, supersoluble, monomial
Aliases: C32⋊2Dic9, C33.2Dic3, C32⋊C9⋊3C4, (C3×C6).3D9, C6.6(C9⋊S3), (C3×C18).6S3, (C3×C9)⋊4Dic3, (C32×C6).3S3, C3.3(C9⋊Dic3), C2.(C32⋊2D9), C6.1(He3⋊C2), C3.1(He3⋊3C4), C32.8(C3⋊Dic3), (C3×C6).16(C3⋊S3), (C2×C32⋊C9).3C2, SmallGroup(324,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — C32⋊2Dic9 |
Generators and relations for C32⋊2Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 233 in 61 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C32, C32, C32, Dic3, C12, C18, C3×C6, C3×C6, C3×C6, C3×C9, C33, Dic9, C3×Dic3, C3⋊Dic3, C3×C18, C32×C6, C32⋊C9, C3×Dic9, C3×C3⋊Dic3, C2×C32⋊C9, C32⋊2Dic9
Quotients: C1, C2, C4, S3, Dic3, D9, C3⋊S3, Dic9, C3⋊Dic3, C9⋊S3, He3⋊C2, C9⋊Dic3, He3⋊3C4, C32⋊2D9, C32⋊2Dic9
►On 36 points
(2 8 14)(3 15 9)(5 11 17)(6 18 12)(20 32 26)(21 27 33)(23 35 29)(24 30 36)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 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)
(1 28 10 19)(2 27 11 36)(3 26 12 35)(4 25 13 34)(5 24 14 33)(6 23 15 32)(7 22 16 31)(8 21 17 30)(9 20 18 29)
G:=sub<Sym(36)| (2,8,14)(3,15,9)(5,11,17)(6,18,12)(20,32,26)(21,27,33)(23,35,29)(24,30,36), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (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), (1,28,10,19)(2,27,11,36)(3,26,12,35)(4,25,13,34)(5,24,14,33)(6,23,15,32)(7,22,16,31)(8,21,17,30)(9,20,18,29)>;
G:=Group( (2,8,14)(3,15,9)(5,11,17)(6,18,12)(20,32,26)(21,27,33)(23,35,29)(24,30,36), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (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), (1,28,10,19)(2,27,11,36)(3,26,12,35)(4,25,13,34)(5,24,14,33)(6,23,15,32)(7,22,16,31)(8,21,17,30)(9,20,18,29) );
G=PermutationGroup([[(2,8,14),(3,15,9),(5,11,17),(6,18,12),(20,32,26),(21,27,33),(23,35,29),(24,30,36)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36)], [(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)], [(1,28,10,19),(2,27,11,36),(3,26,12,35),(4,25,13,34),(5,24,14,33),(6,23,15,32),(7,22,16,31),(8,21,17,30),(9,20,18,29)]])
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 18A | ··· | 18I |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 27 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | ··· | 6 | 27 | 27 | 27 | 27 | 6 | ··· | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | - | - | + | - | |||||
| image | C1 | C2 | C4 | S3 | S3 | Dic3 | Dic3 | D9 | Dic9 | He3⋊C2 | He3⋊3C4 | C32⋊2D9 | C32⋊2Dic9 |
| kernel | C32⋊2Dic9 | C2×C32⋊C9 | C32⋊C9 | C3×C18 | C32×C6 | C3×C9 | C33 | C3×C6 | C32 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 3 | 1 | 3 | 1 | 9 | 9 | 4 | 4 | 2 | 2 |
Matrix representation of C32⋊2Dic9 ►in GL5(𝔽37)
| 14 | 23 | 0 | 0 | 0 |
| 23 | 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 10 |
| 15 | 16 | 0 | 0 | 0 |
| 16 | 27 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 22 | 21 | 0 | 0 | 0 |
| 14 | 15 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 |
| 0 | 0 | 0 | 31 | 0 |
G:=sub<GL(5,GF(37))| [14,23,0,0,0,23,22,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[15,16,0,0,0,16,27,0,0,0,0,0,36,0,0,0,0,0,27,0,0,0,0,0,11],[22,14,0,0,0,21,15,0,0,0,0,0,31,0,0,0,0,0,0,31,0,0,0,31,0] >;
C32⋊2Dic9 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2{\rm Dic}_9 % in TeX
G:=Group("C3^2:2Dic9"); // GroupNames label
G:=SmallGroup(324,20);
// by ID
G=gap.SmallGroup(324,20);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,12,794,338,579,735,2164]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations