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G = C6.D6⋊S3  order 432 = 24·33

3rd semidirect product of C6.D6 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C6.D6⋊3S3, C32⋊2Q8⋊6S3, C33⋊7(C4○D4), C33⋊8D4⋊6C2, Dic3.4(S32), C33⋊9D4⋊10C2, C3⋊Dic3.33D6, C3⋊3(D6.D6), C3⋊1(D6.6D6), C32⋊9(C4○D12), (C3×Dic3).22D6, C32⋊5(Q8⋊3S3), (C32×C6).19C23, (C32×Dic3).7C22, C2.19(S33), C6.19(C2×S32), (C2×C3⋊S3).19D6, C33⋊8(C2×C4)⋊4C2, (C3×C6.D6)⋊3C2, (C3×C32⋊2Q8)⋊7C2, (C6×C3⋊S3).23C22, (C3×C6).68(C22×S3), (C3×C3⋊Dic3).16C22, (C2×C33⋊C2).6C22, SmallGroup(432,612)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C6.D6⋊S3
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C3×C6.D6 — C6.D6⋊S3
Lower central
C33 — C32×C6 — C6.D6⋊S3
Upper central
C1 — C2

Generators and relations for C6.D6⋊S3
 G = < a,b,c,d,e | a6=c2=d3=e2=1, b6=a3, bab-1=cac=eae=a-1, ad=da, cbc=b5, bd=db, be=eb, cd=dc, ece=a3c, ede=d-1 >

Subgroups: 1476 in 218 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C3 [×4], C4 [×4], C22 [×3], S3 [×15], C6, C6 [×2], C6 [×6], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×2], C32 [×4], Dic3, Dic3 [×2], Dic3 [×3], C12 [×10], D6 [×13], C2×C6 [×2], C4○D4, C3×S3 [×6], C3⋊S3 [×11], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×2], C4×S3 [×8], D12 [×7], C3⋊D4 [×4], C2×C12 [×2], C3×Q8, C33, C3×Dic3 [×6], C3×Dic3 [×6], C3⋊Dic3, C3×C12 [×3], S3×C6 [×6], C2×C3⋊S3 [×2], C2×C3⋊S3 [×7], C4○D12 [×2], Q8⋊3S3, C3×C3⋊S3 [×2], C33⋊C2, C32×C6, C6.D6 [×2], C6.D6 [×3], D6⋊S3, C3⋊D12 [×8], C32⋊2Q8, C3×Dic6 [×2], S3×C12 [×4], C4×C3⋊S3, C12⋊S3 [×2], C32×Dic3, C32×Dic3 [×2], C3×C3⋊Dic3, C6×C3⋊S3 [×2], C2×C33⋊C2, D6.D6, D6.6D6 [×2], C3×C6.D6 [×2], C3×C32⋊2Q8, C33⋊8(C2×C4), C33⋊8D4 [×2], C33⋊9D4, C6.D6⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12 [×2], Q8⋊3S3, C2×S32 [×3], D6.D6, D6.6D6 [×2], S33, C6.D6⋊S3

Permutation representations of C6.D6⋊S3
►On 24 points - transitive group 24T1300
Generators in S24
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 24)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 15)(2 20)(3 13)(4 18)(5 23)(6 16)(7 21)(8 14)(9 19)(10 24)(11 17)(12 22)

(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)

(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)
 
G:=sub<Sym(24)| (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)>;
 
G:=Group( (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21) );
 
G=PermutationGroup([(1,11,9,7,5,3),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,16,18,20,22,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,15),(2,20),(3,13),(4,18),(5,23),(6,16),(7,21),(8,14),(9,19),(10,24),(11,17),(12,22)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21)])
 
G:=TransitiveGroup(24,1300);
 

45 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K12A···12H12I···12P12Q
order122223333333444446666666666612···1212···1212
size1118185422244483366182224448181818186···612···1236

45 irreducible representations

dim11111122222224444488
type+++++++++++++++++
imageC1C2C2C2C2C2S3S3D6D6D6C4○D4C4○D12S32Q8⋊3S3C2×S32D6.D6D6.6D6S33C6.D6⋊S3
kernelC6.D6⋊S3C3×C6.D6C3×C32⋊2Q8C33⋊8(C2×C4)C33⋊8D4C33⋊9D4C6.D6C32⋊2Q8C3×Dic3C3⋊Dic3C2×C3⋊S3C33C32Dic3C32C6C3C3C2C1
# reps12112121612283132411

Matrix representation of C6.D6⋊S3 ►in GL8(ℤ)

01000000
-11000000
00010000
00-110000
00000100
0000-1100
00000001
000000-11
,
00000010
0000001-1
00001000
00001-100
001-10000
000-10000
1-1000000
0-1000000
,
001-10000
000-10000
1-1000000
0-1000000
000000-10
000000-11
0000-1000
0000-1100
,
-11000000
-10000000
000-10000
001-10000
0000-1100
0000-1000
0000000-1
0000001-1
,
00000-100
0000-1000
00000001
00000010
0-1000000
-10000000
00010000
00100000

G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0],[-1,-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,-1,0,0,0,0,0,0,0,0,-1,-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,-1],[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,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0] >;
 

C6.D6⋊S3 in GAP, Magma, Sage, TeX 

C_6.D_6\rtimes S_3
 
% in TeX
 
G:=Group("C6.D6:S3");
 
// GroupNames label
 
G:=SmallGroup(432,612);
 
// by ID
 
G=gap.SmallGroup(432,612);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,135,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=c^2=d^3=e^2=1,b^6=a^3,b*a*b^-1=c*a*c=e*a*e=a^-1,a*d=d*a,c*b*c=b^5,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=d^-1>;
 
// generators/relations
 

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