Copied to
clipboard

G = C2×C3⋊D12order 144 = 24·32

Direct product of C2 and C3⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C3⋊D12
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C3⋊D12 — C2×C3⋊D12
 Lower central C32 — C3×C6 — C2×C3⋊D12
 Upper central C1 — C22

Generators and relations for C2×C3⋊D12
G = < a,b,c,d | a2=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 464 in 124 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C3×Dic3, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, C2×C3⋊D4, C3⋊D12, C6×Dic3, S3×C2×C6, C22×C3⋊S3, C2×C3⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C2×C3⋊D12

Character table of C2×C3⋊D12

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D size 1 1 1 1 6 6 18 18 2 2 4 6 6 2 2 2 2 2 2 4 4 4 6 6 6 6 6 6 6 6 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 -2 -2 0 0 2 -1 -1 0 0 2 -1 -1 2 2 -1 -1 -1 -1 1 1 1 1 0 0 0 0 orthogonal lifted from D6 ρ10 2 2 2 2 0 0 0 0 -1 2 -1 -2 -2 -1 2 2 -1 -1 2 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ11 2 -2 -2 2 0 0 0 0 -1 2 -1 2 -2 1 -2 -2 1 -1 2 -1 1 1 0 0 0 0 1 1 -1 -1 orthogonal lifted from D6 ρ12 2 -2 -2 2 0 0 0 0 -1 2 -1 -2 2 1 -2 -2 1 -1 2 -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 -1 2 2 -1 2 2 -1 -1 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 2 2 2 0 0 2 -1 -1 0 0 2 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ15 2 -2 -2 2 -2 2 0 0 2 -1 -1 0 0 -2 1 1 -2 2 -1 -1 1 1 -1 1 1 -1 0 0 0 0 orthogonal lifted from D6 ρ16 2 -2 2 -2 0 0 0 0 2 2 2 0 0 -2 -2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 -2 0 0 0 0 2 2 2 0 0 2 2 -2 -2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 -2 2 2 -2 0 0 2 -1 -1 0 0 -2 1 1 -2 2 -1 -1 1 1 1 -1 -1 1 0 0 0 0 orthogonal lifted from D6 ρ19 2 2 -2 -2 0 0 0 0 -1 2 -1 0 0 -1 2 -2 1 1 -2 1 -1 1 0 0 0 0 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ20 2 -2 2 -2 0 0 0 0 -1 2 -1 0 0 1 -2 2 -1 1 -2 1 1 -1 0 0 0 0 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ21 2 2 -2 -2 0 0 0 0 -1 2 -1 0 0 -1 2 -2 1 1 -2 1 -1 1 0 0 0 0 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ22 2 -2 2 -2 0 0 0 0 -1 2 -1 0 0 1 -2 2 -1 1 -2 1 1 -1 0 0 0 0 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ23 2 2 -2 -2 0 0 0 0 2 -1 -1 0 0 2 -1 1 -2 -2 1 1 -1 1 -√-3 -√-3 √-3 √-3 0 0 0 0 complex lifted from C3⋊D4 ρ24 2 2 -2 -2 0 0 0 0 2 -1 -1 0 0 2 -1 1 -2 -2 1 1 -1 1 √-3 √-3 -√-3 -√-3 0 0 0 0 complex lifted from C3⋊D4 ρ25 2 -2 2 -2 0 0 0 0 2 -1 -1 0 0 -2 1 -1 2 -2 1 1 1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 complex lifted from C3⋊D4 ρ26 2 -2 2 -2 0 0 0 0 2 -1 -1 0 0 -2 1 -1 2 -2 1 1 1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 complex lifted from C3⋊D4 ρ27 4 -4 4 -4 0 0 0 0 -2 -2 1 0 0 2 2 -2 -2 2 2 -1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ28 4 4 4 4 0 0 0 0 -2 -2 1 0 0 -2 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ29 4 -4 -4 4 0 0 0 0 -2 -2 1 0 0 2 2 2 2 -2 -2 1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ30 4 4 -4 -4 0 0 0 0 -2 -2 1 0 0 -2 -2 2 2 2 2 -1 1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C3⋊D12

Permutation representations of C2×C3⋊D12
On 24 points - transitive group 24T230
Generators in S24
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 17 21)(14 22 18)(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 16)(2 15)(3 14)(4 13)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)

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

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

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

G:=TransitiveGroup(24,230);

Matrix representation of C2×C3⋊D12 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0
,
 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C3⋊D12 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{12}
% in TeX

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

G:=SmallGroup(144,151);
// by ID

G=gap.SmallGroup(144,151);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,55,490,3461]);
// Polycyclic

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

Export

׿
×
𝔽