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## G = (C3×C6).8D12order 432 = 24·33

### 1st non-split extension by C3×C6 of D12 acting via D12/C3=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — (C3×C6).8D12
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×C32⋊4D6 — (C3×C6).8D12
 Lower central C33 — C3×C3⋊S3 — (C3×C6).8D12
 Upper central C1 — C2

Generators and relations for (C3×C6).8D12
G = < a,b,c,d | a3=b6=c12=1, d2=b3, ab=ba, cac-1=b2, dad-1=a-1, cbc-1=ab3, bd=db, dcd-1=b3c-1 >

Subgroups: 1024 in 132 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3⋊Dic3, C32⋊C4, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, D6⋊C4, C3×C3⋊S3, C3×C3⋊S3, C32×C6, S3×Dic3, C6.D6, C2×C32⋊C4, C2×S32, C3×C3⋊Dic3, C3×C32⋊C4, C324D6, C324D6, C6×C3⋊S3, C6×C3⋊S3, S32⋊C4, C339(C2×C4), C6×C32⋊C4, C2×C324D6, (C3×C6).8D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, D6⋊C4, S3≀C2, S32⋊C4, C322D12, (C3×C6).8D12

Character table of (C3×C6).8D12

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E 12F size 1 1 9 9 18 18 2 4 4 8 8 18 18 18 18 2 4 4 8 8 18 18 36 36 18 18 18 18 36 36 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 i i -i -i -1 -1 -1 -1 -1 -1 1 1 -1 i i -i -i -i i linear of order 4 ρ6 1 -1 1 -1 1 -1 1 1 1 1 1 -i -i i i -1 -1 -1 -1 -1 -1 1 1 -1 -i -i i i i -i linear of order 4 ρ7 1 -1 1 -1 -1 1 1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 -1 1 -1 1 i i -i -i i -i linear of order 4 ρ8 1 -1 1 -1 -1 1 1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 -1 1 -1 1 -i -i i i -i i linear of order 4 ρ9 2 2 2 2 0 0 -1 2 2 -1 -1 0 2 2 0 -1 2 2 -1 -1 -1 -1 0 0 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ10 2 2 -2 -2 0 0 2 2 2 2 2 0 0 0 0 2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 0 -1 2 2 -1 -1 0 -2 -2 0 -1 2 2 -1 -1 -1 -1 0 0 1 1 1 1 0 0 orthogonal lifted from D6 ρ13 2 2 -2 -2 0 0 -1 2 2 -1 -1 0 0 0 0 -1 2 2 -1 -1 1 1 0 0 √3 -√3 -√3 √3 0 0 orthogonal lifted from D12 ρ14 2 2 -2 -2 0 0 -1 2 2 -1 -1 0 0 0 0 -1 2 2 -1 -1 1 1 0 0 -√3 √3 √3 -√3 0 0 orthogonal lifted from D12 ρ15 2 -2 2 -2 0 0 -1 2 2 -1 -1 0 2i -2i 0 1 -2 -2 1 1 1 -1 0 0 -i -i i i 0 0 complex lifted from C4×S3 ρ16 2 -2 2 -2 0 0 -1 2 2 -1 -1 0 -2i 2i 0 1 -2 -2 1 1 1 -1 0 0 i i -i -i 0 0 complex lifted from C4×S3 ρ17 2 -2 -2 2 0 0 -1 2 2 -1 -1 0 0 0 0 1 -2 -2 1 1 -1 1 0 0 √-3 -√-3 √-3 -√-3 0 0 complex lifted from C3⋊D4 ρ18 2 -2 -2 2 0 0 -1 2 2 -1 -1 0 0 0 0 1 -2 -2 1 1 -1 1 0 0 -√-3 √-3 -√-3 √-3 0 0 complex lifted from C3⋊D4 ρ19 4 -4 0 0 2 -2 4 -2 1 -2 1 0 0 0 0 -4 -1 2 -1 2 0 0 -1 1 0 0 0 0 0 0 orthogonal lifted from S32⋊C4 ρ20 4 4 0 0 0 0 4 1 -2 1 -2 2 0 0 2 4 -2 1 -2 1 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from S3≀C2 ρ21 4 -4 0 0 -2 2 4 -2 1 -2 1 0 0 0 0 -4 -1 2 -1 2 0 0 1 -1 0 0 0 0 0 0 orthogonal lifted from S32⋊C4 ρ22 4 4 0 0 -2 -2 4 -2 1 -2 1 0 0 0 0 4 1 -2 1 -2 0 0 1 1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ23 4 4 0 0 2 2 4 -2 1 -2 1 0 0 0 0 4 1 -2 1 -2 0 0 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ24 4 4 0 0 0 0 4 1 -2 1 -2 -2 0 0 -2 4 -2 1 -2 1 0 0 0 0 0 0 0 0 1 1 orthogonal lifted from S3≀C2 ρ25 4 -4 0 0 0 0 4 1 -2 1 -2 -2i 0 0 2i -4 2 -1 2 -1 0 0 0 0 0 0 0 0 -i i complex lifted from S32⋊C4 ρ26 4 -4 0 0 0 0 4 1 -2 1 -2 2i 0 0 -2i -4 2 -1 2 -1 0 0 0 0 0 0 0 0 i -i complex lifted from S32⋊C4 ρ27 8 -8 0 0 0 0 -4 -4 2 2 -1 0 0 0 0 4 -2 4 1 -2 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ28 8 8 0 0 0 0 -4 -4 2 2 -1 0 0 0 0 -4 2 -4 -1 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ29 8 8 0 0 0 0 -4 2 -4 -1 2 0 0 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12 ρ30 8 -8 0 0 0 0 -4 2 -4 -1 2 0 0 0 0 4 4 -2 -2 1 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of (C3×C6).8D12
On 24 points - transitive group 24T1305
Generators in S24
(1 9 5)(2 6 10)(3 7 11)(4 12 8)(13 17 21)(14 22 18)(15 23 19)(16 20 24)
(1 23 5 15 9 19)(2 24 6 16 10 20)(3 21 11 17 7 13)(4 22 12 18 8 14)
(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 12 15 14)(2 13 16 11)(3 10 17 24)(4 23 18 9)(5 8 19 22)(6 21 20 7)

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

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

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

G:=TransitiveGroup(24,1305);

Matrix representation of (C3×C6).8D12 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 1 12 0 0 0 1 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 1 0 0 0 0 12 0 0 0 0 1 12 0 0 0 12 0 0 0
,
 11 11 0 0 0 0 2 9 0 0 0 0 0 0 0 0 12 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 12 0 0
,
 11 4 0 0 0 0 2 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12,12,0,0,0,12,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,12,0,0,0,0,1,0,0,0,0,12,12,0,0,0,1,0,0,0],[11,2,0,0,0,0,11,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,0,12,0],[11,2,0,0,0,0,4,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C3×C6).8D12 in GAP, Magma, Sage, TeX

(C_3\times C_6)._8D_{12}
% in TeX

G:=Group("(C3xC6).8D12");
// GroupNames label

G:=SmallGroup(432,586);
// by ID

G=gap.SmallGroup(432,586);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,92,1684,1691,298,677,348,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^12=1,d^2=b^3,a*b=b*a,c*a*c^-1=b^2,d*a*d^-1=a^-1,c*b*c^-1=a*b^3,b*d=d*b,d*c*d^-1=b^3*c^-1>;
// generators/relations

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