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## G = S3×C4○D4order 96 = 25·3

### Direct product of S3 and C4○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C4○D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4 — S3×C4○D4
 Lower central C3 — C6 — S3×C4○D4
 Upper central C1 — C4 — C4○D4

Generators and relations for S3×C4○D4
G = < a,b,c,d,e | a3=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 330 in 164 conjugacy classes, 87 normal (16 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], S3 [×2], S3 [×3], C6, C6 [×3], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], Dic3, Dic3 [×3], C12, C12 [×3], D6, D6 [×3], D6 [×6], C2×C6 [×3], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], Dic6 [×3], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C2×C4○D4, S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, S3×C4○D4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, S3×C23, S3×C4○D4

Character table of S3×C4○D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 12A 12B 12C 12D 12E size 1 1 2 2 2 3 3 6 6 6 2 1 1 2 2 2 3 3 6 6 6 2 4 4 4 2 2 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 0 0 0 0 0 -1 -2 -2 2 -2 -2 0 0 0 0 0 -1 1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ18 2 2 2 -2 -2 0 0 0 0 0 -1 2 2 -2 2 -2 0 0 0 0 0 -1 1 -1 1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ19 2 2 2 2 -2 0 0 0 0 0 -1 -2 -2 -2 -2 2 0 0 0 0 0 -1 -1 -1 1 1 1 1 1 -1 orthogonal lifted from D6 ρ20 2 2 2 2 2 0 0 0 0 0 -1 2 2 2 2 2 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ21 2 2 -2 -2 -2 0 0 0 0 0 -1 -2 -2 2 2 2 0 0 0 0 0 -1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ22 2 2 -2 -2 2 0 0 0 0 0 -1 2 2 -2 -2 2 0 0 0 0 0 -1 1 1 -1 -1 -1 1 1 -1 orthogonal lifted from D6 ρ23 2 2 -2 2 2 0 0 0 0 0 -1 -2 -2 -2 2 -2 0 0 0 0 0 -1 -1 1 -1 1 1 1 -1 1 orthogonal lifted from D6 ρ24 2 2 -2 2 -2 0 0 0 0 0 -1 2 2 2 -2 -2 0 0 0 0 0 -1 -1 1 1 -1 -1 -1 1 1 orthogonal lifted from D6 ρ25 2 -2 0 0 0 2 -2 0 0 0 2 2i -2i 0 0 0 2i -2i 0 0 0 -2 0 0 0 2i -2i 0 0 0 complex lifted from C4○D4 ρ26 2 -2 0 0 0 2 -2 0 0 0 2 -2i 2i 0 0 0 -2i 2i 0 0 0 -2 0 0 0 -2i 2i 0 0 0 complex lifted from C4○D4 ρ27 2 -2 0 0 0 -2 2 0 0 0 2 2i -2i 0 0 0 -2i 2i 0 0 0 -2 0 0 0 2i -2i 0 0 0 complex lifted from C4○D4 ρ28 2 -2 0 0 0 -2 2 0 0 0 2 -2i 2i 0 0 0 2i -2i 0 0 0 -2 0 0 0 -2i 2i 0 0 0 complex lifted from C4○D4 ρ29 4 -4 0 0 0 0 0 0 0 0 -2 -4i 4i 0 0 0 0 0 0 0 0 2 0 0 0 2i -2i 0 0 0 complex faithful ρ30 4 -4 0 0 0 0 0 0 0 0 -2 4i -4i 0 0 0 0 0 0 0 0 2 0 0 0 -2i 2i 0 0 0 complex faithful

Permutation representations of S3×C4○D4
On 24 points - transitive group 24T101
Generators in S24
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 13)(10 24 14)(11 21 15)(12 22 16)
(1 3)(2 4)(5 20)(6 17)(7 18)(8 19)(9 11)(10 12)(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 2 3 4)(5 6 7 8)(9 12 11 10)(13 16 15 14)(17 18 19 20)(21 24 23 22)
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 17)(14 18)(15 19)(16 20)

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

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

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

G:=TransitiveGroup(24,101);

Matrix representation of S3×C4○D4 in GL4(𝔽5) generated by

 4 0 3 0 0 4 0 2 3 0 0 0 0 2 0 0
,
 1 0 0 0 0 0 0 3 2 0 4 0 0 2 0 0
,
 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3
,
 2 0 0 0 0 3 0 0 0 0 2 0 0 0 0 3
,
 0 2 0 1 0 0 4 0 0 4 0 0 1 0 2 0
G:=sub<GL(4,GF(5))| [4,0,3,0,0,4,0,2,3,0,0,0,0,2,0,0],[1,0,2,0,0,0,0,2,0,0,4,0,0,3,0,0],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[0,0,0,1,2,0,4,0,0,4,0,2,1,0,0,0] >;

S3×C4○D4 in GAP, Magma, Sage, TeX

S_3\times C_4\circ D_4
% in TeX

G:=Group("S3xC4oD4");
// GroupNames label

G:=SmallGroup(96,215);
// by ID

G=gap.SmallGroup(96,215);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,297,2309]);
// Polycyclic

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

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