Copied to
clipboard

## G = C4×C32⋊C6order 216 = 23·33

### Direct product of C4 and C32⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×C32⋊C6
 Chief series C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C4×C32⋊C6
 Lower central C32 — C4×C32⋊C6
 Upper central C1 — C4

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, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C32⋊C6, C2×He3, S3×C12, C4×C3⋊S3, C32⋊C12, C4×He3, C2×C32⋊C6, C4×C32⋊C6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, C3×S3, C4×S3, C2×C12, S3×C6, C32⋊C6, S3×C12, C2×C32⋊C6, C4×C32⋊C6

Smallest permutation representation of C4×C32⋊C6
On 36 points
Generators in S36
(1 4 11 9)(2 5 12 7)(3 6 10 8)(13 28 19 35)(14 29 20 36)(15 30 21 31)(16 25 22 32)(17 26 23 33)(18 27 24 34)
(2 17 14)(3 18 15)(5 26 29)(6 27 30)(7 33 36)(8 34 31)(10 24 21)(12 23 20)
(1 13 16)(2 17 14)(3 15 18)(4 28 25)(5 26 29)(6 30 27)(7 33 36)(8 31 34)(9 35 32)(10 21 24)(11 19 22)(12 23 20)
(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)| (1,4,11,9)(2,5,12,7)(3,6,10,8)(13,28,19,35)(14,29,20,36)(15,30,21,31)(16,25,22,32)(17,26,23,33)(18,27,24,34), (2,17,14)(3,18,15)(5,26,29)(6,27,30)(7,33,36)(8,34,31)(10,24,21)(12,23,20), (1,13,16)(2,17,14)(3,15,18)(4,28,25)(5,26,29)(6,30,27)(7,33,36)(8,31,34)(9,35,32)(10,21,24)(11,19,22)(12,23,20), (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( (1,4,11,9)(2,5,12,7)(3,6,10,8)(13,28,19,35)(14,29,20,36)(15,30,21,31)(16,25,22,32)(17,26,23,33)(18,27,24,34), (2,17,14)(3,18,15)(5,26,29)(6,27,30)(7,33,36)(8,34,31)(10,24,21)(12,23,20), (1,13,16)(2,17,14)(3,15,18)(4,28,25)(5,26,29)(6,30,27)(7,33,36)(8,31,34)(9,35,32)(10,21,24)(11,19,22)(12,23,20), (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([[(1,4,11,9),(2,5,12,7),(3,6,10,8),(13,28,19,35),(14,29,20,36),(15,30,21,31),(16,25,22,32),(17,26,23,33),(18,27,24,34)], [(2,17,14),(3,18,15),(5,26,29),(6,27,30),(7,33,36),(8,34,31),(10,24,21),(12,23,20)], [(1,13,16),(2,17,14),(3,15,18),(4,28,25),(5,26,29),(6,30,27),(7,33,36),(8,31,34),(9,35,32),(10,21,24),(11,19,22),(12,23,20)], [(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)  He35M4(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
He35M4(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

׿
×
𝔽