non-abelian, supersoluble, monomial
Aliases: C32⋊C6⋊C4, C3⋊S3⋊Dic3, He3⋊2(C2×C4), (C3×C6).2D6, C6.17(S32), C3⋊Dic3⋊1S3, C32⋊1(C4×S3), C32⋊(C2×Dic3), C32⋊C12⋊2C2, He3⋊3C4⋊2C2, C3.2(S3×Dic3), C2.1(C32⋊D6), (C2×He3).2C22, (C2×C3⋊S3).S3, (C2×C32⋊C6).C2, SmallGroup(216,34)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C32⋊C6⋊C4 |
Generators and relations for C32⋊C6⋊C4
G = < a,b,c,d | a3=b3=c6=d4=1, dad-1=ab=ba, cac-1=a-1b-1, cbc-1=dbd-1=b-1, dcd-1=c-1 >
Subgroups: 282 in 62 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×2], C22, S3 [×4], C6, C6 [×5], C2×C4, C32 [×2], C32, Dic3 [×5], C12 [×2], D6 [×2], C2×C6, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C3×C6, C4×S3 [×2], C2×Dic3, He3, C3×Dic3 [×4], C3⋊Dic3, S3×C6, C2×C3⋊S3, C32⋊C6 [×2], C2×He3, S3×Dic3, C6.D6, C32⋊C12, He3⋊3C4, C2×C32⋊C6, C32⋊C6⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, Dic3 [×2], D6 [×2], C4×S3, C2×Dic3, S32, S3×Dic3, C32⋊D6, C32⋊C6⋊C4
Character table of C32⋊C6⋊C4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 9 | 9 | 2 | 6 | 6 | 12 | 9 | 9 | 9 | 9 | 2 | 6 | 6 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | -i | i | i | -1 | -1 | -1 | -1 | 1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | i | -i | -i | -1 | -1 | -1 | -1 | 1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | -2 | 0 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 2i | 0 | -2i | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | -i | i | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | -2i | 0 | 2i | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | i | -i | complex lifted from C4×S3 |
| ρ17 | 4 | 4 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ18 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Dic3, Schur index 2 |
| ρ19 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | -2 | 0 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from C32⋊D6 |
| ρ20 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from C32⋊D6 |
| ρ21 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 3 | 0 | 0 | 0 | 0 | 0 | i | -i | 0 | 0 | complex faithful |
| ρ22 | 6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 3 | 0 | 0 | 0 | 0 | 0 | -i | i | 0 | 0 | complex faithful |
►On 36 points
(1 28 19)(2 29 20)(4 22 25)(5 23 26)(7 17 32)(8 18 33)(10 35 14)(11 36 15)
(1 28 19)(2 20 29)(3 30 21)(4 22 25)(5 26 23)(6 24 27)(7 17 32)(8 33 18)(9 13 34)(10 35 14)(11 15 36)(12 31 16)
(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 10 4 7)(2 9 5 12)(3 8 6 11)(13 23 31 29)(14 22 32 28)(15 21 33 27)(16 20 34 26)(17 19 35 25)(18 24 36 30)
G:=sub<Sym(36)| (1,28,19)(2,29,20)(4,22,25)(5,23,26)(7,17,32)(8,18,33)(10,35,14)(11,36,15), (1,28,19)(2,20,29)(3,30,21)(4,22,25)(5,26,23)(6,24,27)(7,17,32)(8,33,18)(9,13,34)(10,35,14)(11,15,36)(12,31,16), (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), (1,10,4,7)(2,9,5,12)(3,8,6,11)(13,23,31,29)(14,22,32,28)(15,21,33,27)(16,20,34,26)(17,19,35,25)(18,24,36,30)>;
G:=Group( (1,28,19)(2,29,20)(4,22,25)(5,23,26)(7,17,32)(8,18,33)(10,35,14)(11,36,15), (1,28,19)(2,20,29)(3,30,21)(4,22,25)(5,26,23)(6,24,27)(7,17,32)(8,33,18)(9,13,34)(10,35,14)(11,15,36)(12,31,16), (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), (1,10,4,7)(2,9,5,12)(3,8,6,11)(13,23,31,29)(14,22,32,28)(15,21,33,27)(16,20,34,26)(17,19,35,25)(18,24,36,30) );
G=PermutationGroup([(1,28,19),(2,29,20),(4,22,25),(5,23,26),(7,17,32),(8,18,33),(10,35,14),(11,36,15)], [(1,28,19),(2,20,29),(3,30,21),(4,22,25),(5,26,23),(6,24,27),(7,17,32),(8,33,18),(9,13,34),(10,35,14),(11,15,36),(12,31,16)], [(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)], [(1,10,4,7),(2,9,5,12),(3,8,6,11),(13,23,31,29),(14,22,32,28),(15,21,33,27),(16,20,34,26),(17,19,35,25),(18,24,36,30)])
Matrix representation of C32⋊C6⋊C4 ►in GL6(𝔽13)
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[0,0,0,0,0,12,0,0,0,0,12,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8,0,0,0] >;
C32⋊C6⋊C4 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_6\rtimes C_4
% in TeX
G:=Group("C3^2:C6:C4"); // GroupNames label
G:=SmallGroup(216,34);
// by ID
G=gap.SmallGroup(216,34);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,201,1444,382,5189,2603]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^6=d^4=1,d*a*d^-1=a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations