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G = C2×C32.A4order 216 = 23·33

Direct product of C2 and C32.A4

Aliases: C2×C32.A4, C62.7C6, C2323- 1+2, C3.A43C6, C6.6(C3×A4), C3.4(C6×A4), (C3×C6).3A4, C32.(C2×A4), (C2×C62).2C3, (C22×C6).4C32, C222(C2×3- 1+2), (C2×C3.A4)⋊2C3, (C2×C6).4(C3×C6), SmallGroup(216,106)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C32.A4
 Chief series C1 — C22 — C2×C6 — C62 — C32.A4 — C2×C32.A4
 Lower central C22 — C2×C6 — C2×C32.A4
 Upper central C1 — C6 — C3×C6

Generators and relations for C2×C32.A4
G = < a,b,c,d,e,f | a2=b3=c3=d2=e2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 136 in 56 conjugacy classes, 20 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3, C22, C22 [×2], C6, C6 [×9], C23, C9 [×3], C32, C2×C6, C2×C6 [×9], C18 [×3], C3×C6, C3×C6 [×2], C22×C6, C22×C6, 3- 1+2, C3.A4 [×3], C62, C62 [×2], C2×3- 1+2, C2×C3.A4 [×3], C2×C62, C32.A4, C2×C32.A4
Quotients: C1, C2, C3 [×4], C6 [×4], C32, A4, C3×C6, C2×A4, 3- 1+2, C3×A4, C2×3- 1+2, C6×A4, C32.A4, C2×C32.A4

Permutation representations of C2×C32.A4
On 18 points - transitive group 18T92
Generators in S18
(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(2 17)(3 18)(5 11)(6 12)(8 14)(9 15)
(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,92);

C2×C32.A4 is a maximal subgroup of   C62.Dic3
C2×C32.A4 is a maximal quotient of   Q8⋊C94C6

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C ··· 6T 9A ··· 9F 18A ··· 18F order 1 2 2 2 3 3 3 3 6 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 3 3 1 1 3 3 1 1 3 ··· 3 12 ··· 12 12 ··· 12

40 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 A4 C2×A4 3- 1+2 C3×A4 C2×3- 1+2 C6×A4 C32.A4 C2×C32.A4 kernel C2×C32.A4 C32.A4 C2×C3.A4 C2×C62 C3.A4 C62 C3×C6 C32 C23 C6 C22 C3 C2 C1 # reps 1 1 6 2 6 2 1 1 2 2 2 2 6 6

Matrix representation of C2×C32.A4 in GL3(𝔽7) generated by

 6 0 0 0 6 0 0 0 6
,
 4 0 0 0 2 0 0 0 1
,
 4 0 0 0 4 0 0 0 4
,
 1 0 0 0 6 0 0 0 6
,
 6 0 0 0 6 0 0 0 1
,
 0 6 0 0 0 3 1 0 0
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,0,0,0,2,0,0,0,1],[4,0,0,0,4,0,0,0,4],[1,0,0,0,6,0,0,0,6],[6,0,0,0,6,0,0,0,1],[0,0,1,6,0,0,0,3,0] >;

C2×C32.A4 in GAP, Magma, Sage, TeX

C_2\times C_3^2.A_4
% in TeX

G:=Group("C2xC3^2.A4");
// GroupNames label

G:=SmallGroup(216,106);
// by ID

G=gap.SmallGroup(216,106);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,115,224,1630,2927]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^2=e^2=1,f^3=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c^-1,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations

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