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## G = C2×S3×D4order 96 = 25·3

### Direct product of C2, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×S3×D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C2×S3×D4
 Lower central C3 — C6 — C2×S3×D4
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×S3×D4
G = < a,b,c,d,e | a2=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 562 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×D4, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C2×S3×D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, C2×S3×D4

Character table of C2×S3×D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 1 1 1 2 2 2 2 3 3 3 3 6 6 6 6 2 2 2 6 6 2 2 2 4 4 4 4 4 4 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 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 -1 2 2 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ18 2 -2 -2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 2 0 0 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 -1 -2 -2 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ21 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 -1 -2 2 0 0 1 1 -1 -1 1 1 -1 -1 1 orthogonal lifted from D6 ρ22 2 2 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 -1 2 -2 0 0 1 1 -1 1 -1 1 -1 1 -1 orthogonal lifted from D6 ρ23 2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 -1 -2 2 0 0 1 1 -1 1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ24 2 2 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -1 2 -2 0 0 1 1 -1 -1 1 -1 1 1 -1 orthogonal lifted from D6 ρ25 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ26 2 -2 2 -2 0 0 0 0 2 -2 2 -2 0 0 0 0 2 0 0 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 -2 -2 2 0 0 0 0 -2 -2 2 2 0 0 0 0 2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 2 2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 -1 -2 -2 0 0 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ29 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ30 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 -2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4

Permutation representations of C2×S3×D4
On 24 points - transitive group 24T143
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 21)(10 22)(11 23)(12 24)(13 20)(14 17)(15 18)(16 19)
(1 21 20)(2 22 17)(3 23 18)(4 24 19)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 8)(2 5)(3 6)(4 7)(9 20)(10 17)(11 18)(12 19)(13 21)(14 22)(15 23)(16 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 8)(2 7)(3 6)(4 5)(9 21)(10 24)(11 23)(12 22)(13 20)(14 19)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(24,143);

Matrix representation of C2×S3×D4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 0 12 0 0 1 12 0 0 0 0 1 0 0 0 0 1
,
 0 12 0 0 12 0 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 1 10 0 0 5 12
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 5 12
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,5,0,0,10,12],[1,0,0,0,0,1,0,0,0,0,1,5,0,0,0,12] >;

C2×S3×D4 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_4
% in TeX

G:=Group("C2xS3xD4");
// GroupNames label

G:=SmallGroup(96,209);
// by ID

G=gap.SmallGroup(96,209);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,159,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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