direct product, metabelian, supersoluble, monomial
Aliases: C4×C32⋊C6, C3⋊S3⋊C12, (C3×C12)⋊3S3, (C3×C12)⋊2C6, He3⋊4(C2×C4), (C3×C6).7D6, C6.10(S3×C6), C3.2(S3×C12), (C4×He3)⋊3C2, C3⋊Dic3⋊2C6, C32⋊3(C4×S3), C12.12(C3×S3), C32⋊C12⋊5C2, C32⋊1(C2×C12), (C2×He3).7C22, (C4×C3⋊S3)⋊C3, (C2×C3⋊S3).C6, (C3×C6).2(C2×C6), C2.1(C2×C32⋊C6), (C2×C32⋊C6).2C2, SmallGroup(216,50)
Series: Derived ►Chief ►Lower central ►Upper central
C32 — C4×C32⋊C6 |
Generators and relations for C4×C32⋊C6
G = < a,b,c,d | a4=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >
Subgroups: 232 in 62 conjugacy classes, 25 normal (21 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C4, C22, S3 [×4], C6, C6 [×5], C2×C4, C32 [×2], C32, Dic3 [×2], C12, C12 [×4], D6 [×2], C2×C6, C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×2], C3×C6, C4×S3 [×2], C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12 [×2], C3×C12, S3×C6, C2×C3⋊S3, C32⋊C6 [×2], C2×He3, S3×C12, C4×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C4×C32⋊C6
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, C12 [×2], D6, C2×C6, C3×S3, C4×S3, C2×C12, S3×C6, C32⋊C6, S3×C12, C2×C32⋊C6, C4×C32⋊C6
(1 10 6 9)(2 11 4 7)(3 12 5 8)(13 21 30 31)(14 22 25 32)(15 23 26 33)(16 24 27 34)(17 19 28 35)(18 20 29 36)
(2 28 25)(3 29 26)(4 17 14)(5 18 15)(7 19 22)(8 20 23)(11 35 32)(12 36 33)
(1 30 27)(2 28 25)(3 26 29)(4 17 14)(5 15 18)(6 13 16)(7 19 22)(8 23 20)(9 21 24)(10 31 34)(11 35 32)(12 33 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)| (1,10,6,9)(2,11,4,7)(3,12,5,8)(13,21,30,31)(14,22,25,32)(15,23,26,33)(16,24,27,34)(17,19,28,35)(18,20,29,36), (2,28,25)(3,29,26)(4,17,14)(5,18,15)(7,19,22)(8,20,23)(11,35,32)(12,36,33), (1,30,27)(2,28,25)(3,26,29)(4,17,14)(5,15,18)(6,13,16)(7,19,22)(8,23,20)(9,21,24)(10,31,34)(11,35,32)(12,33,36), (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( (1,10,6,9)(2,11,4,7)(3,12,5,8)(13,21,30,31)(14,22,25,32)(15,23,26,33)(16,24,27,34)(17,19,28,35)(18,20,29,36), (2,28,25)(3,29,26)(4,17,14)(5,18,15)(7,19,22)(8,20,23)(11,35,32)(12,36,33), (1,30,27)(2,28,25)(3,26,29)(4,17,14)(5,15,18)(6,13,16)(7,19,22)(8,23,20)(9,21,24)(10,31,34)(11,35,32)(12,33,36), (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([(1,10,6,9),(2,11,4,7),(3,12,5,8),(13,21,30,31),(14,22,25,32),(15,23,26,33),(16,24,27,34),(17,19,28,35),(18,20,29,36)], [(2,28,25),(3,29,26),(4,17,14),(5,18,15),(7,19,22),(8,20,23),(11,35,32),(12,36,33)], [(1,30,27),(2,28,25),(3,26,29),(4,17,14),(5,15,18),(6,13,16),(7,19,22),(8,23,20),(9,21,24),(10,31,34),(11,35,32),(12,33,36)], [(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)])
C4×C32⋊C6 is a maximal subgroup of
C32⋊C6⋊C8 He3⋊M4(2) He3⋊5M4(2) C3⋊S3⋊Dic6 C12⋊S3⋊S3 C12.91S32 C12.S32 C3⋊S3⋊D12 C62.36D6 C62.13D6 (Q8×He3)⋊C2
C4×C32⋊C6 is a maximal quotient of
He3⋊5M4(2) C62.19D6 C62.21D6
40 conjugacy classes
class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12L | 12M | 12N | 12O | 12P |
order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
size | 1 | 1 | 9 | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 1 | 9 | 9 | 2 | 3 | 3 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 9 | 9 | 9 | 9 |
40 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
type | + | + | + | + | + | + | + | + | |||||||||||
image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C12 | C32⋊C6 | C2×C32⋊C6 | C4×C32⋊C6 |
kernel | C4×C32⋊C6 | C32⋊C12 | C4×He3 | C2×C32⋊C6 | C4×C3⋊S3 | C32⋊C6 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | C3⋊S3 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 | C4 | C2 | C1 |
# reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C4×C32⋊C6 ►in GL8(𝔽13)
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,GF(13))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
C4×C32⋊C6 in GAP, Magma, Sage, TeX
C_4\times C_3^2\rtimes C_6
% in TeX
G:=Group("C4xC3^2:C6");
// GroupNames label
G:=SmallGroup(216,50);
// by ID
G=gap.SmallGroup(216,50);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,79,1444,736,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;
// generators/relations