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G = C2×C322D12order 432 = 24·33

Direct product of C2 and C322D12

direct product, non-abelian, soluble, monomial

Aliases: C2×C322D12, C61S3≀C2, C3⋊S32D12, (C3×C6)⋊2D12, C32⋊C42D6, C334(C2×D4), (C32×C6)⋊3D4, C323(C2×D12), C324D63C22, C32(C2×S3≀C2), (C3×C3⋊S3)⋊6D4, (C6×C32⋊C4)⋊5C2, (C2×C32⋊C4)⋊3S3, (C2×C3⋊S3).22D6, (C3×C3⋊S3).7C23, C3⋊S3.4(C22×S3), (C3×C32⋊C4)⋊2C22, (C6×C3⋊S3).35C22, (C2×C324D6)⋊6C2, SmallGroup(432,756)

Series: Derived Chief Lower central Upper central

C1C32C3×C3⋊S3 — C2×C322D12
C1C3C33C3×C3⋊S3C324D6C322D12 — C2×C322D12
C33C3×C3⋊S3 — C2×C322D12
C1C2

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 12A2B2C2D2E2F2G3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K12A12B12C12D
 size 1199181818182448818182448818183636363618181818
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-1-111111111111111-1-1-1-11111    linear of order 2
ρ31-11-11-11-1111111-1-1-1-1-1-1-111-11-1-11-11    linear of order 2
ρ41-11-1-11-11111111-1-1-1-1-1-1-11-11-11-11-11    linear of order 2
ρ51-11-111-1-111111-11-1-1-1-1-1-1111-1-11-11-1    linear of order 2
ρ61-11-1-1-11111111-11-1-1-1-1-1-11-1-1111-11-1    linear of order 2
ρ711111-1-1111111-1-111111111-1-11-1-1-1-1    linear of order 2
ρ81111-111-111111-1-11111111-111-1-1-1-1-1    linear of order 2
ρ92-22-20000-122-1-1-221-2-2111-10000-11-11    orthogonal lifted from D6
ρ102-2-2200002222200-2-2-2-2-22-200000000    orthogonal lifted from D4
ρ1122220000-122-1-122-122-1-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ122-22-20000-122-1-12-21-2-2111-100001-11-1    orthogonal lifted from D6
ρ1322220000-122-1-1-2-2-122-1-1-1-100001111    orthogonal lifted from D6
ρ1422-2-20000222220022222-2-200000000    orthogonal lifted from D4
ρ1522-2-20000-122-1-100-122-1-111000033-3-3    orthogonal lifted from D12
ρ1622-2-20000-122-1-100-122-1-1110000-3-333    orthogonal lifted from D12
ρ172-2-220000-122-1-1001-2-211-110000-333-3    orthogonal lifted from D12
ρ182-2-220000-122-1-1001-2-211-1100003-3-33    orthogonal lifted from D12
ρ194-400200-241-2-2100-4-12-1200-10010000    orthogonal lifted from C2×S3≀C2
ρ204-4000-2204-211-200-42-12-10001-100000    orthogonal lifted from C2×S3≀C2
ρ2144000-2-204-211-2004-21-210001100000    orthogonal lifted from S3≀C2
ρ224400200241-2-210041-21-200-100-10000    orthogonal lifted from S3≀C2
ρ234-40002-204-211-200-42-12-1000-1100000    orthogonal lifted from C2×S3≀C2
ρ24440002204-211-2004-21-21000-1-100000    orthogonal lifted from S3≀C2
ρ254400-200-241-2-210041-21-20010010000    orthogonal lifted from S3≀C2
ρ264-400-200241-2-2100-4-12-1200100-10000    orthogonal lifted from C2×S3≀C2
ρ278-8000000-4-42-120044-2-210000000000    orthogonal faithful
ρ2888000000-42-42-100-42-4-120000000000    orthogonal lifted from C322D12
ρ298-8000000-42-42-1004-241-20000000000    orthogonal faithful
ρ3088000000-4-42-1200-4-422-10000000000    orthogonal lifted from C322D12

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)

1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012001
000100
000010
0012000
,
100000
010000
001000
0001210
0001200
000001
,
3100000
360000
000100
000001
001000
000010
,
1030000
630000
0000012
0001200
0000120
0012000

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

Export

Character table of C2×C322D12 in TeX

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