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## G = S3×C8⋊C22order 192 = 26·3

### Direct product of S3 and C8⋊C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — S3×C8⋊C22
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×D4 — S3×C8⋊C22
 Lower central C3 — C6 — C12 — S3×C8⋊C22
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for S3×C8⋊C22
G = < a,b,c,d,e | a3=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >

Subgroups: 944 in 298 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×10], C3, C4 [×2], C4 [×4], C22, C22 [×24], S3 [×2], S3 [×4], C6, C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4, D4 [×2], D4 [×14], Q8, Q8 [×2], C23 [×12], Dic3 [×2], Dic3, C12 [×2], C12, D6 [×2], D6 [×17], C2×C6, C2×C6 [×5], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×2], D8 [×6], SD16 [×2], SD16 [×6], C22×C4 [×2], C2×D4, C2×D4 [×10], C2×Q8, C4○D4, C4○D4 [×5], C24, C3⋊C8 [×2], C24 [×2], Dic6, Dic6, C4×S3 [×4], C4×S3 [×3], D12, D12 [×2], D12 [×2], C2×Dic3, C2×Dic3, C3⋊D4 [×7], C2×C12, C2×C12, C3×D4, C3×D4 [×2], C3×D4 [×2], C3×Q8, C22×S3, C22×S3 [×10], C22×C6, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22, C8⋊C22 [×7], C22×D4, C2×C4○D4, S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24 [×2], C4.Dic3, D4⋊S3 [×4], D4.S3 [×2], Q82S3 [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, C4○D12, S3×D4, S3×D4 [×4], S3×D4 [×3], D42S3, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×C23, C2×C8⋊C22, S3×M4(2), C8⋊D6, S3×D8 [×2], D8⋊S3 [×2], S3×SD16 [×2], Q83D6 [×2], D126C22, D4⋊D6, C3×C8⋊C22, C2×S3×D4, S3×C4○D4, S3×C8⋊C22
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C8⋊C22 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C8⋊C22, C2×S3×D4, S3×C8⋊C22

Permutation representations of S3×C8⋊C22
On 24 points - transitive group 24T362
Generators in S24
(1 23 9)(2 24 10)(3 17 11)(4 18 12)(5 19 13)(6 20 14)(7 21 15)(8 22 16)
(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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 3)(2 6)(5 7)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)

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

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

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

G:=TransitiveGroup(24,362);

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 24A 24B order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 24 24 size 1 1 2 3 3 4 4 4 6 12 12 12 2 2 2 4 6 6 12 2 4 8 8 8 4 4 12 12 4 4 8 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 D6 D6 C8⋊C22 S3×D4 S3×D4 S3×C8⋊C22 kernel S3×C8⋊C22 S3×M4(2) C8⋊D6 S3×D8 D8⋊S3 S3×SD16 Q8⋊3D6 D12⋊6C22 D4⋊D6 C3×C8⋊C22 C2×S3×D4 S3×C4○D4 C8⋊C22 C4×S3 C2×Dic3 C22×S3 M4(2) D8 SD16 C2×D4 C4○D4 S3 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 1 2 1 1 1 2 2 1 1 2 1 1 1

Matrix representation of S3×C8⋊C22 in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1],[0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

S3×C8⋊C22 in GAP, Magma, Sage, TeX

S_3\times C_8\rtimes C_2^2
% in TeX

G:=Group("S3xC8:C2^2");
// GroupNames label

G:=SmallGroup(192,1331);
// by ID

G=gap.SmallGroup(192,1331);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,570,185,438,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^8=d^2=e^2=1,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,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
// generators/relations

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