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G = C3⋊S3⋊D12  order 432 = 24·33

The semidirect product of C3⋊S3 and D12 acting via D12/C4=S3

non-abelian, supersoluble, monomial

Aliases: C3⋊S3⋊D12, C12.24S32, (C3×C12)⋊1D6, He3⋊2(C2×D4), C3⋊Dic3⋊4D6, C32⋊C6⋊1D4, C3.3(S3×D12), C12⋊S3⋊5S3, C32⋊1(S3×D4), He3⋊5D4⋊4C2, He3⋊4D4⋊6C2, He3⋊3D4⋊1C2, C4⋊1(C32⋊D6), C32⋊1(C2×D12), (C4×He3)⋊1C22, C32⋊C12⋊3C22, (C2×He3).8C23, (C4×C3⋊S3)⋊3S3, C6.82(C2×S32), (C2×C3⋊S3)⋊1D6, (C4×C32⋊C6)⋊4C2, (C2×C32⋊D6)⋊1C2, (C3×C6).8(C22×S3), C2.10(C2×C32⋊D6), (C2×C32⋊C6)⋊1C22, (C2×He3⋊C2)⋊1C22, SmallGroup(432,301)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — C3⋊S3⋊D12
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C2×C32⋊C6 — C2×C32⋊D6 — C3⋊S3⋊D12
Lower central
He3 — C2×He3 — C3⋊S3⋊D12
Upper central
C1 — C2 — C4

Generators and relations for C3⋊S3⋊D12
 G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=eae=a-1, ad=da, cbc=b-1, dbd-1=ebe=a-1b, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1571 in 205 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×D12, S3×D4, C32⋊C6, C32⋊C6, He3⋊C2, C2×He3, D6⋊S3, C3⋊D12, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, C2×S32, C32⋊C12, C4×He3, C32⋊D6, C2×C32⋊C6, C2×C32⋊C6, C2×He3⋊C2, S3×D12, D6⋊D6, He3⋊3D4, C4×C32⋊C6, He3⋊4D4, He3⋊5D4, C2×C32⋊D6, C3⋊S3⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, S32, C2×D12, S3×D4, C2×S32, C32⋊D6, S3×D12, C2×C32⋊D6, C3⋊S3⋊D12

Smallest permutation representation of C3⋊S3⋊D12
►On 36 points
Generators in S36
(1 30 23)(2 31 24)(3 32 13)(4 33 14)(5 34 15)(6 35 16)(7 36 17)(8 25 18)(9 26 19)(10 27 20)(11 28 21)(12 29 22)

(2 24 31)(3 32 13)(5 15 34)(6 35 16)(8 18 25)(9 26 19)(11 21 28)(12 29 22)

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)

(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 3)(4 12)(5 11)(6 10)(7 9)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)
 
G:=sub<Sym(36)| (1,30,23)(2,31,24)(3,32,13)(4,33,14)(5,34,15)(6,35,16)(7,36,17)(8,25,18)(9,26,19)(10,27,20)(11,28,21)(12,29,22), (2,24,31)(3,32,13)(5,15,34)(6,35,16)(8,18,25)(9,26,19)(11,21,28)(12,29,22), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)>;
 
G:=Group( (1,30,23)(2,31,24)(3,32,13)(4,33,14)(5,34,15)(6,35,16)(7,36,17)(8,25,18)(9,26,19)(10,27,20)(11,28,21)(12,29,22), (2,24,31)(3,32,13)(5,15,34)(6,35,16)(8,18,25)(9,26,19)(11,21,28)(12,29,22), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,25), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31) );
 
G=PermutationGroup([[(1,30,23),(2,31,24),(3,32,13),(4,33,14),(5,34,15),(6,35,16),(7,36,17),(8,25,18),(9,26,19),(10,27,20),(11,28,21),(12,29,22)], [(2,24,31),(3,32,13),(5,15,34),(6,35,16),(8,18,25),(9,26,19),(11,21,28),(12,29,22)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,25)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31)]])
 

32 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D4A4B6A6B6C6D6E6F6G6H6I6J12A12B12C12D12E12F12G12H
order1222222233334466666666661212121212121212
size11991818181826612218266121818363636364661212121818

32 irreducible representations

dim111111122222222444466
type++++++++++++++++++++
imageC1C2C2C2C2C2C3⋊S3⋊D12S3S3D4D6D6D6D12S32S3×D4C2×S32S3×D12C32⋊D6C2×C32⋊D6
kernelC3⋊S3⋊D12He3⋊3D4C4×C32⋊C6He3⋊4D4He3⋊5D4C2×C32⋊D6C1C4×C3⋊S3C12⋊S3C32⋊C6C3⋊Dic3C3×C12C2×C3⋊S3C3⋊S3C12C32C6C3C4C2
# reps12111211121234111222

Matrix representation of C3⋊S3⋊D12 ►in GL10(𝔽13)

1000000000
0100000000
0010000000
0001000000
00001210000
00001200000
00000012100
00000012000
0000101001
00000120121212
,
0100000000
121200000000
0001000000
001212000000
0000100000
0000010000
00000001200
00000011200
0000000101
000012121201212
,
12000000000
1100000000
00120000000
0011000000
0000010000
0000100000
0000000100
0000001000
0000000010
0000121212121212
,
90110000000
09011000000
2040000000
0204000000
00000000121
0000121212121112
0000100000
0000010000
0000000010
0000001010
,
12000000000
01200000000
4010000000
0401000000
0000121212121112
00000000121
0000000100
0000001000
0000000010
0000010010

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,1,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,12,12,1,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,12,12,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12],[12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,12],[9,0,2,0,0,0,0,0,0,0,0,9,0,2,0,0,0,0,0,0,11,0,4,0,0,0,0,0,0,0,0,11,0,4,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,11,0,0,1,1,0,0,0,0,1,12,0,0,0,0],[12,0,4,0,0,0,0,0,0,0,0,12,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12,0,0,1,0,0,0,0,0,0,12,0,1,0,0,0,0,0,0,0,11,12,0,0,1,1,0,0,0,0,12,1,0,0,0,0] >;
 

C3⋊S3⋊D12 in GAP, Magma, Sage, TeX 

C_3\rtimes S_3\rtimes D_{12}
 
% in TeX
 
G:=Group("C3:S3:D12");
 
// GroupNames label
 
G:=SmallGroup(432,301);
 
// by ID
 
G=gap.SmallGroup(432,301);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,a*b=b*a,c*a*c=e*a*e=a^-1,a*d=d*a,c*b*c=b^-1,d*b*d^-1=e*b*e=a^-1*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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