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## G = C2×C32⋊4D6order 216 = 23·33

### Direct product of C2 and C32⋊4D6

Aliases: C2×C324D6, C334C23, C62S32, C3⋊S33D6, (C3×C6)⋊5D6, C326(C22×S3), (C32×C6)⋊3C22, C33(C2×S32), (C2×C3⋊S3)⋊7S3, (C6×C3⋊S3)⋊9C2, (C3×C3⋊S3)⋊4C22, SmallGroup(216,172)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C2×C32⋊4D6
 Chief series C1 — C3 — C32 — C33 — C3×C3⋊S3 — C32⋊4D6 — C2×C32⋊4D6
 Lower central C33 — C2×C32⋊4D6
 Upper central C1 — C2

Generators and relations for C2×C324D6
G = < a,b,c,d,e | a2=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 772 in 162 conjugacy classes, 39 normal (5 characteristic)
C1, C2, C2 [×6], C3 [×3], C3 [×4], C22 [×7], S3 [×18], C6 [×3], C6 [×10], C23, C32 [×3], C32 [×4], D6 [×21], C2×C6 [×3], C3×S3 [×18], C3⋊S3 [×6], C3×C6 [×3], C3×C6 [×4], C22×S3 [×3], C33, S32 [×12], S3×C6 [×9], C2×C3⋊S3 [×3], C3×C3⋊S3 [×6], C32×C6, C2×S32 [×3], C324D6 [×4], C6×C3⋊S3 [×3], C2×C324D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C22×S3 [×3], S32 [×3], C2×S32 [×3], C324D6, C2×C324D6

Character table of C2×C324D6

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N size 1 1 9 9 9 9 9 9 2 2 2 4 4 4 4 4 2 2 2 4 4 4 4 4 18 18 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ3 1 -1 1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ4 1 -1 -1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ5 1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 linear of order 2 ρ6 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 linear of order 2 ρ7 1 -1 -1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 -1 linear of order 2 ρ8 1 -1 1 1 -1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ9 2 2 0 0 0 2 2 0 2 2 -1 -1 -1 2 -1 -1 2 -1 2 -1 -1 -1 2 -1 0 -1 0 0 -1 0 orthogonal lifted from S3 ρ10 2 -2 0 0 0 2 -2 0 2 2 -1 -1 -1 2 -1 -1 -2 1 -2 1 1 1 -2 1 0 1 0 0 -1 0 orthogonal lifted from D6 ρ11 2 -2 2 0 0 0 0 -2 -1 2 2 -1 2 -1 -1 -1 -2 -2 1 1 1 1 1 -2 0 0 0 1 0 -1 orthogonal lifted from D6 ρ12 2 2 -2 0 0 0 0 -2 -1 2 2 -1 2 -1 -1 -1 2 2 -1 -1 -1 -1 -1 2 0 0 0 1 0 1 orthogonal lifted from D6 ρ13 2 -2 0 0 0 -2 2 0 2 2 -1 -1 -1 2 -1 -1 -2 1 -2 1 1 1 -2 1 0 -1 0 0 1 0 orthogonal lifted from D6 ρ14 2 2 0 0 0 -2 -2 0 2 2 -1 -1 -1 2 -1 -1 2 -1 2 -1 -1 -1 2 -1 0 1 0 0 1 0 orthogonal lifted from D6 ρ15 2 -2 0 -2 2 0 0 0 2 -1 2 -1 -1 -1 2 -1 1 -2 -2 -2 1 1 1 1 -1 0 1 0 0 0 orthogonal lifted from D6 ρ16 2 2 0 2 2 0 0 0 2 -1 2 -1 -1 -1 2 -1 -1 2 2 2 -1 -1 -1 -1 -1 0 -1 0 0 0 orthogonal lifted from S3 ρ17 2 -2 -2 0 0 0 0 2 -1 2 2 -1 2 -1 -1 -1 -2 -2 1 1 1 1 1 -2 0 0 0 -1 0 1 orthogonal lifted from D6 ρ18 2 -2 0 2 -2 0 0 0 2 -1 2 -1 -1 -1 2 -1 1 -2 -2 -2 1 1 1 1 1 0 -1 0 0 0 orthogonal lifted from D6 ρ19 2 2 0 -2 -2 0 0 0 2 -1 2 -1 -1 -1 2 -1 -1 2 2 2 -1 -1 -1 -1 1 0 1 0 0 0 orthogonal lifted from D6 ρ20 2 2 2 0 0 0 0 2 -1 2 2 -1 2 -1 -1 -1 2 2 -1 -1 -1 -1 -1 2 0 0 0 -1 0 -1 orthogonal lifted from S3 ρ21 4 -4 0 0 0 0 0 0 4 -2 -2 1 1 -2 -2 1 2 2 -4 2 -1 -1 2 -1 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ22 4 -4 0 0 0 0 0 0 -2 -2 4 1 -2 1 -2 1 2 -4 2 2 -1 -1 -1 2 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ23 4 4 0 0 0 0 0 0 -2 -2 4 1 -2 1 -2 1 -2 4 -2 -2 1 1 1 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ24 4 4 0 0 0 0 0 0 -2 4 -2 1 -2 -2 1 1 4 -2 -2 1 1 1 -2 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ25 4 4 0 0 0 0 0 0 4 -2 -2 1 1 -2 -2 1 -2 -2 4 -2 1 1 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ26 4 -4 0 0 0 0 0 0 -2 4 -2 1 -2 -2 1 1 -4 2 2 -1 -1 -1 2 2 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ27 4 -4 0 0 0 0 0 0 -2 -2 -2 -1-3√-3/2 1 1 1 -1+3√-3/2 2 2 2 -1 1-3√-3/2 1+3√-3/2 -1 -1 0 0 0 0 0 0 complex faithful ρ28 4 4 0 0 0 0 0 0 -2 -2 -2 -1-3√-3/2 1 1 1 -1+3√-3/2 -2 -2 -2 1 -1+3√-3/2 -1-3√-3/2 1 1 0 0 0 0 0 0 complex lifted from C32⋊4D6 ρ29 4 4 0 0 0 0 0 0 -2 -2 -2 -1+3√-3/2 1 1 1 -1-3√-3/2 -2 -2 -2 1 -1-3√-3/2 -1+3√-3/2 1 1 0 0 0 0 0 0 complex lifted from C32⋊4D6 ρ30 4 -4 0 0 0 0 0 0 -2 -2 -2 -1+3√-3/2 1 1 1 -1-3√-3/2 2 2 2 -1 1+3√-3/2 1-3√-3/2 -1 -1 0 0 0 0 0 0 complex faithful

Permutation representations of C2×C324D6
On 24 points - transitive group 24T548
Generators in S24
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)(13 15 17)(14 18 16)(19 21 23)(20 24 22)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)(13 15 17)(14 18 16)(19 23 21)(20 22 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)
(1 21)(2 20)(3 19)(4 24)(5 23)(6 22)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)

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

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

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

G:=TransitiveGroup(24,548);

C2×C324D6 is a maximal subgroup of
C3⋊S3.2D12  (C3×C6).8D12  Dic36S32  D6⋊S32  C3⋊S34D12  C123S32  C6224D6  C2×S33
C2×C324D6 is a maximal quotient of
C3⋊S34Dic6  C12⋊S312S3  C12.95S32  C123S32  C62.96D6  C6224D6

Matrix representation of C2×C324D6 in GL4(𝔽7) generated by

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

C2×C324D6 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_4D_6
% in TeX

G:=Group("C2xC3^2:4D6");
// GroupNames label

G:=SmallGroup(216,172);
// by ID

G=gap.SmallGroup(216,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,387,201,730,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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