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## G = C6×C32⋊C4order 216 = 23·33

### Direct product of C6 and C32⋊C4

Aliases: C6×C32⋊C4, C3⋊S33C12, (C3×C6)⋊2C12, C332(C2×C4), (C32×C6)⋊1C4, C323(C2×C12), (C3×C3⋊S3)⋊2C4, (C6×C3⋊S3).4C2, (C2×C3⋊S3).3C6, C3⋊S3.3(C2×C6), (C3×C3⋊S3).7C22, SmallGroup(216,168)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C3×C3⋊S3 — C3×C32⋊C4 — C6×C32⋊C4
 Lower central C32 — C6×C32⋊C4
 Upper central C1 — C6

Generators and relations for C6×C32⋊C4
G = < a,b,c,d | a6=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 248 in 60 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4 [×2], C22, S3 [×4], C6, C6 [×6], C2×C4, C32, C32 [×4], C12 [×2], D6 [×2], C2×C6, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×4], C2×C12, C33, C32⋊C4 [×2], S3×C6 [×2], C2×C3⋊S3, C3×C3⋊S3 [×2], C32×C6, C2×C32⋊C4, C3×C32⋊C4 [×2], C6×C3⋊S3, C6×C32⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, C12 [×2], C2×C6, C2×C12, C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C6×C32⋊C4

Permutation representations of C6×C32⋊C4
On 24 points - transitive group 24T555
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(7 9 11)(8 10 12)(19 23 21)(20 24 22)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)
(1 8 16 24)(2 9 17 19)(3 10 18 20)(4 11 13 21)(5 12 14 22)(6 7 15 23)

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

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

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

G:=TransitiveGroup(24,555);

C6×C32⋊C4 is a maximal subgroup of   D6⋊(C32⋊C4)  C33⋊(C4⋊C4)  (C3×C6).8D12  (C3×C6).9D12  C6.PSU3(𝔽2)  C6.2PSU3(𝔽2)

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I 6J 6K 6L 12A ··· 12H order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 9 9 1 1 4 ··· 4 9 9 9 9 1 1 4 ··· 4 9 9 9 9 9 ··· 9

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C6×C32⋊C4 kernel C6×C32⋊C4 C3×C32⋊C4 C6×C3⋊S3 C2×C32⋊C4 C3×C3⋊S3 C32×C6 C32⋊C4 C2×C3⋊S3 C3⋊S3 C3×C6 C6 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 2 4 4

Matrix representation of C6×C32⋊C4 in GL4(𝔽7) generated by

 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 0 0 4 6 2 3 1 0 6 1 2 2 4 4 5 0
,
 4 2 5 0 2 3 5 2 2 2 6 2 2 4 3 2
,
 5 0 5 3 6 4 6 4 5 4 6 6 1 6 3 6
G:=sub<GL(4,GF(7))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,2,6,4,0,3,1,4,4,1,2,5,6,0,2,0],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[5,6,5,1,0,4,4,6,5,6,6,3,3,4,6,6] >;

C6×C32⋊C4 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes C_4
% in TeX

G:=Group("C6xC3^2:C4");
// GroupNames label

G:=SmallGroup(216,168);
// by ID

G=gap.SmallGroup(216,168);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,3,72,5044,142,6917,455]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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