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## G = C2×C32⋊2D12order 432 = 24·33

### Direct product of C2 and C32⋊2D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — C2×C32⋊2D12
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C32⋊4D6 — C32⋊2D12 — C2×C32⋊2D12
 Lower central C33 — C3×C3⋊S3 — C2×C32⋊2D12
 Upper central C1 — C2

Generators and relations for C2×C322D12
G = < a,b,c,d,e | a2=b3=c3=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 1616 in 192 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2 [×6], C3, C3 [×4], C4 [×2], C22 [×9], S3 [×16], C6, C6 [×10], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], C12 [×2], D6 [×22], C2×C6 [×3], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3 [×4], C3×C6, C3×C6 [×4], D12 [×4], C2×C12, C22×S3 [×4], C33, C32⋊C4 [×2], S32 [×14], S3×C6 [×8], C2×C3⋊S3, C2×C3⋊S3 [×2], C2×D12, C3×C3⋊S3 [×2], C3×C3⋊S3 [×4], C32×C6, S3≀C2 [×4], C2×C32⋊C4, C2×S32 [×4], C3×C32⋊C4 [×2], C324D6 [×4], C324D6 [×2], C6×C3⋊S3, C6×C3⋊S3 [×2], C2×S3≀C2, C322D12 [×4], C6×C32⋊C4, C2×C324D6 [×2], C2×C322D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D12 [×2], C22×S3, C2×D12, S3≀C2, C2×S3≀C2, C322D12, C2×C322D12

Character table of C2×C322D12

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 12A 12B 12C 12D size 1 1 9 9 18 18 18 18 2 4 4 8 8 18 18 2 4 4 8 8 18 18 36 36 36 36 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 2 -2 0 0 0 0 -1 2 2 -1 -1 -2 2 1 -2 -2 1 1 1 -1 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 0 2 2 2 2 2 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 0 0 -1 2 2 -1 -1 2 2 -1 2 2 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 2 -2 0 0 0 0 -1 2 2 -1 -1 2 -2 1 -2 -2 1 1 1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ13 2 2 2 2 0 0 0 0 -1 2 2 -1 -1 -2 -2 -1 2 2 -1 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 -2 0 0 0 0 2 2 2 2 2 0 0 2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 0 0 0 -1 2 2 -1 -1 0 0 -1 2 2 -1 -1 1 1 0 0 0 0 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ16 2 2 -2 -2 0 0 0 0 -1 2 2 -1 -1 0 0 -1 2 2 -1 -1 1 1 0 0 0 0 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ17 2 -2 -2 2 0 0 0 0 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 -1 1 0 0 0 0 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ18 2 -2 -2 2 0 0 0 0 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 -1 1 0 0 0 0 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ19 4 -4 0 0 2 0 0 -2 4 1 -2 -2 1 0 0 -4 -1 2 -1 2 0 0 -1 0 0 1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ20 4 -4 0 0 0 -2 2 0 4 -2 1 1 -2 0 0 -4 2 -1 2 -1 0 0 0 1 -1 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ21 4 4 0 0 0 -2 -2 0 4 -2 1 1 -2 0 0 4 -2 1 -2 1 0 0 0 1 1 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ22 4 4 0 0 2 0 0 2 4 1 -2 -2 1 0 0 4 1 -2 1 -2 0 0 -1 0 0 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ23 4 -4 0 0 0 2 -2 0 4 -2 1 1 -2 0 0 -4 2 -1 2 -1 0 0 0 -1 1 0 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ24 4 4 0 0 0 2 2 0 4 -2 1 1 -2 0 0 4 -2 1 -2 1 0 0 0 -1 -1 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ25 4 4 0 0 -2 0 0 -2 4 1 -2 -2 1 0 0 4 1 -2 1 -2 0 0 1 0 0 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ26 4 -4 0 0 -2 0 0 2 4 1 -2 -2 1 0 0 -4 -1 2 -1 2 0 0 1 0 0 -1 0 0 0 0 orthogonal lifted from C2×S3≀C2 ρ27 8 -8 0 0 0 0 0 0 -4 -4 2 -1 2 0 0 4 4 -2 -2 1 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ28 8 8 0 0 0 0 0 0 -4 2 -4 2 -1 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 0 0 -4 2 -4 2 -1 0 0 4 -2 4 1 -2 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ30 8 8 0 0 0 0 0 0 -4 -4 2 -1 2 0 0 -4 -4 2 2 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊2D12

Permutation representations of C2×C322D12
On 24 points - transitive group 24T1304
Generators in S24
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 13)(9 14)(10 15)(11 16)(12 17)
(1 9 5)(2 6 10)(3 7 11)(4 12 8)(13 21 17)(14 22 18)(15 19 23)(16 20 24)
(1 5 9)(2 6 10)(3 11 7)(4 12 8)(13 21 17)(14 18 22)(15 19 23)(16 24 20)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)

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

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

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

G:=TransitiveGroup(24,1304);

Matrix representation of C2×C322D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 12 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1
,
 3 10 0 0 0 0 3 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0
,
 10 3 0 0 0 0 6 3 0 0 0 0 0 0 0 0 0 12 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0 0 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1],[3,3,0,0,0,0,10,6,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0],[10,6,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0,0,0] >;

C2×C322D12 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2D_{12}
% in TeX

G:=Group("C2xC3^2:2D12");
// GroupNames label

G:=SmallGroup(432,756);
// by ID

G=gap.SmallGroup(432,756);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,64,1684,1691,165,677,348,530,14118]);
// Polycyclic

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

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