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G = C33⋊6(C2×Q8)  order 432 = 24·33

3rd semidirect product of C33 and C2×Q8 acting via C2×Q8/C2=C23

metabelian, supersoluble, monomial

Aliases: C33⋊6(C2×Q8), Dic3.6S32, C32⋊7(S3×Q8), C32⋊2Q8⋊3S3, C33⋊C2⋊2Q8, C33⋊5Q8⋊4C2, C3⋊Dic3.28D6, (C3×Dic3).9D6, C3⋊2(Dic3.D6), (C32×C6).12C23, (C32×Dic3).3C22, C2.12S33, C6.12(C2×S32), C33⋊8(C2×C4).1C2, (C3×C32⋊2Q8)⋊4C2, (C3×C6).61(C22×S3), (C3×C3⋊Dic3).12C22, (C2×C33⋊C2).3C22, SmallGroup(432,605)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — C33⋊6(C2×Q8)
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×Dic3 — C3×C32⋊2Q8 — C33⋊6(C2×Q8)
Lower central
C33 — C32×C6 — C33⋊6(C2×Q8)
Upper central
C1 — C2

Generators and relations for C33⋊6(C2×Q8)
 G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, ac=ca, dad=eae-1=faf-1=a-1, bc=cb, dbd=b-1, be=eb, bf=fb, dcd=ece-1=c-1, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1300 in 210 conjugacy classes, 48 normal (7 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, Dic3, Dic3, C12, D6, C2×Q8, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8, C33⋊C2, C32×C6, C6.D6, C32⋊2Q8, C32⋊2Q8, C3×Dic6, C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, Dic3.D6, C3×C32⋊2Q8, C33⋊8(C2×C4), C33⋊5Q8, C33⋊6(C2×Q8)
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C22×S3, S32, S3×Q8, C2×S32, Dic3.D6, S33, C33⋊6(C2×Q8)

Permutation representations of C33⋊6(C2×Q8)
►On 24 points - transitive group 24T1302
Generators in S24
(1 10 19)(2 20 11)(3 12 17)(4 18 9)(5 23 14)(6 15 24)(7 21 16)(8 13 22)

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

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

(9 18)(10 19)(11 20)(12 17)(13 22)(14 23)(15 24)(16 21)

(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 7 3 5)(2 6 4 8)(9 13 11 15)(10 16 12 14)(17 23 19 21)(18 22 20 24)
 
G:=sub<Sym(24)| (1,10,19)(2,20,11)(3,12,17)(4,18,9)(5,23,14)(6,15,24)(7,21,16)(8,13,22), (1,10,19)(2,11,20)(3,12,17)(4,9,18)(5,14,23)(6,15,24)(7,16,21)(8,13,22), (1,19,10)(2,11,20)(3,17,12)(4,9,18)(5,23,14)(6,15,24)(7,21,16)(8,13,22), (9,18)(10,19)(11,20)(12,17)(13,22)(14,23)(15,24)(16,21), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,23,19,21)(18,22,20,24)>;
 
G:=Group( (1,10,19)(2,20,11)(3,12,17)(4,18,9)(5,23,14)(6,15,24)(7,21,16)(8,13,22), (1,10,19)(2,11,20)(3,12,17)(4,9,18)(5,14,23)(6,15,24)(7,16,21)(8,13,22), (1,19,10)(2,11,20)(3,17,12)(4,9,18)(5,23,14)(6,15,24)(7,21,16)(8,13,22), (9,18)(10,19)(11,20)(12,17)(13,22)(14,23)(15,24)(16,21), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7,3,5)(2,6,4,8)(9,13,11,15)(10,16,12,14)(17,23,19,21)(18,22,20,24) );
 
G=PermutationGroup([[(1,10,19),(2,20,11),(3,12,17),(4,18,9),(5,23,14),(6,15,24),(7,21,16),(8,13,22)], [(1,10,19),(2,11,20),(3,12,17),(4,9,18),(5,14,23),(6,15,24),(7,16,21),(8,13,22)], [(1,19,10),(2,11,20),(3,17,12),(4,9,18),(5,23,14),(6,15,24),(7,21,16),(8,13,22)], [(9,18),(10,19),(11,20),(12,17),(13,22),(14,23),(15,24),(16,21)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,7,3,5),(2,6,4,8),(9,13,11,15),(10,16,12,14),(17,23,19,21),(18,22,20,24)]])
 
G:=TransitiveGroup(24,1302);
 

39 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F3G4A4B4C4D4E4F6A6B6C6D6E6F6G12A···12L12M12N12O
order12223333333444444666666612···12121212
size1127272224448666181818222444812···12363636

39 irreducible representations

dim11112222444488
type+++++-+++-+++
imageC1C2C2C2S3Q8D6D6S32S3×Q8C2×S32Dic3.D6S33C33⋊6(C2×Q8)
kernelC33⋊6(C2×Q8)C3×C32⋊2Q8C33⋊8(C2×C4)C33⋊5Q8C32⋊2Q8C33⋊C2C3×Dic3C3⋊Dic3Dic3C32C6C3C2C1
# reps13313263333611

Matrix representation of C33⋊6(C2×Q8) ►in GL8(ℤ)

-1-1000000
10000000
00010000
00-1-10000
0000-1-100
00001000
00000001
000000-1-1
,
-1-1000000
10000000
00-1-10000
00100000
00000100
0000-1-100
00000001
000000-1-1
,
01000000
-1-1000000
00010000
00-1-10000
00000100
0000-1-100
00000001
000000-1-1
,
-10000000
11000000
00-100000
00110000
0000-1000
00001100
000000-10
00000011
,
00001000
0000-1-100
00000010
000000-1-1
-10000000
11000000
00-100000
00110000
,
00100000
00010000
-10000000
0-1000000
000000-10
0000000-1
00001000
00000100

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

C33⋊6(C2×Q8) in GAP, Magma, Sage, TeX 

C_3^3\rtimes_6(C_2\times Q_8)
 
% in TeX
 
G:=Group("C3^3:6(C2xQ8)");
 
// GroupNames label
 
G:=SmallGroup(432,605);
 
// by ID
 
G=gap.SmallGroup(432,605);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,64,254,135,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,d*a*d=e*a*e^-1=f*a*f^-1=a^-1,b*c=c*b,d*b*d=b^-1,b*e=e*b,b*f=f*b,d*c*d=e*c*e^-1=c^-1,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

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