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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 [×2], C3 [×3], C3 [×4], C4 [×6], C22, S3 [×14], C6 [×3], C6 [×4], C2×C4 [×3], Q8 [×4], C32 [×3], C32 [×4], Dic3 [×3], Dic3 [×9], C12 [×12], D6 [×7], C2×Q8, C3⋊S3 [×14], C3×C6 [×3], C3×C6 [×4], Dic6 [×9], C4×S3 [×12], C3×Q8 [×3], C33, C3×Dic3 [×6], C3×Dic3 [×12], C3⋊Dic3 [×3], C3×C12 [×3], C2×C3⋊S3 [×7], S3×Q8 [×3], C33⋊C2 [×2], C32×C6, C6.D6 [×9], C322Q8 [×3], C322Q8 [×3], C3×Dic6 [×6], C4×C3⋊S3 [×3], C32×Dic3 [×3], C3×C3⋊Dic3 [×3], C2×C33⋊C2, Dic3.D6 [×3], C3×C322Q8 [×3], C338(C2×C4) [×3], C335Q8, C336(C2×Q8)
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], Q8 [×2], C23, D6 [×9], C2×Q8, C22×S3 [×3], S32 [×3], S3×Q8 [×3], C2×S32 [×3], Dic3.D6 [×3], S33, C336(C2×Q8)

Permutation representations of C336(C2×Q8)
On 24 points - transitive group 24T1302
Generators in S24
(1 5 19)(2 20 6)(3 7 17)(4 18 8)(9 23 14)(10 15 24)(11 21 16)(12 13 22)
(1 5 19)(2 6 20)(3 7 17)(4 8 18)(9 14 23)(10 15 24)(11 16 21)(12 13 22)
(1 19 5)(2 6 20)(3 17 7)(4 8 18)(9 23 14)(10 15 24)(11 21 16)(12 13 22)
(5 19)(6 20)(7 17)(8 18)(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 11 3 9)(2 10 4 12)(5 16 7 14)(6 15 8 13)(17 23 19 21)(18 22 20 24)

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

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

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