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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

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 C336(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, C322Q8, C322Q8, C3×Dic6, C4×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C2×C33⋊C2, Dic3.D6, C3×C322Q8, C338(C2×C4), C335Q8, C336(C2×Q8)
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C22×S3, S32, S3×Q8, C2×S32, Dic3.D6, S33, C336(C2×Q8)

Permutation representations of C336(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 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 12A ··· 12L 12M 12N 12O order 1 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 12 ··· 12 12 12 12 size 1 1 27 27 2 2 2 4 4 4 8 6 6 6 18 18 18 2 2 2 4 4 4 8 12 ··· 12 36 36 36

39 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 4 4 4 8 8 type + + + + + - + + + - + + + image C1 C2 C2 C2 S3 Q8 D6 D6 S32 S3×Q8 C2×S32 Dic3.D6 S33 C33⋊6(C2×Q8) kernel C33⋊6(C2×Q8) C3×C32⋊2Q8 C33⋊8(C2×C4) C33⋊5Q8 C32⋊2Q8 C33⋊C2 C3×Dic3 C3⋊Dic3 Dic3 C32 C6 C3 C2 C1 # reps 1 3 3 1 3 2 6 3 3 3 3 6 1 1

Matrix representation of C336(C2×Q8) in GL8(ℤ)

 -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 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 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

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] >;

C336(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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