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G = S3×C8⋊C22  order 192 = 26·3

Direct product of S3 and C8⋊C22

direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary

Aliases: S3×C8⋊C22, D8⋊3D6, C24⋊C23, SD16⋊1D6, D12⋊3C23, D24⋊1C22, M4(2)⋊7D6, C12.19C24, Dic6⋊3C23, (S3×D8)⋊1C2, C3⋊C8⋊3C23, C4○D4⋊12D6, (C2×D4)⋊29D6, C8⋊D6⋊1C2, Q8⋊3D6⋊1C2, D8⋊S3⋊1C2, C8⋊1(C22×S3), D4⋊D6⋊9C2, (S3×C8)⋊1C22, D4⋊S3⋊5C22, (S3×SD16)⋊1C2, (C4×S3).42D4, D6.67(C2×D4), C4.189(S3×D4), D4⋊3(C22×S3), (S3×D4)⋊8C22, (C3×D4)⋊3C23, (C3×D8)⋊1C22, Q8⋊4(C22×S3), (C3×Q8)⋊3C23, (S3×Q8)⋊9C22, C24⋊C2⋊1C22, C8⋊S3⋊1C22, D12⋊6C22⋊9C2, C12.240(C2×D4), C4○D12⋊7C22, (C6×D4)⋊21C22, (S3×M4(2))⋊1C2, C4.19(S3×C23), D4.S3⋊4C22, C22.46(S3×D4), (C2×D12)⋊35C22, D4⋊2S3⋊9C22, (C4×S3).12C23, (C2×Dic3).81D4, Dic3.59(C2×D4), Q8⋊3S3⋊9C22, Q8⋊2S3⋊4C22, (C3×SD16)⋊1C22, C6.120(C22×D4), (C2×C12).110C23, (C22×S3).101D4, (C3×M4(2))⋊1C22, C4.Dic3⋊12C22, (C2×S3×D4)⋊24C2, C3⋊4(C2×C8⋊C22), C2.93(C2×S3×D4), (S3×C4○D4)⋊3C2, (C3×C8⋊C22)⋊1C2, (C2×C6).65(C2×D4), (C3×C4○D4)⋊5C22, (S3×C2×C4).160C22, (C2×C4).94(C22×S3), Aut(D24), Hol(C24), SmallGroup(192,1331)

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, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C8⋊C22, C22×D4, C2×C4○D4, S3×C8, C8⋊S3, C24⋊C2, D24, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3×M4(2), C3×D8, C3×SD16, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, C4○D12, S3×D4, S3×D4, S3×D4, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×C23, C2×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, S3×C8⋊C22
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C8⋊C22, C22×D4, S3×D4, 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 20 9)(2 21 10)(3 22 11)(4 23 12)(5 24 13)(6 17 14)(7 18 15)(8 19 16)

(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)

(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 21)(18 24)(20 22)

(1 5)(3 7)(9 13)(11 15)(18 22)(20 24)
 
G:=sub<Sym(24)| (1,20,9)(2,21,10)(3,22,11)(4,23,12)(5,24,13)(6,17,14)(7,18,15)(8,19,16), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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)(17,21)(18,24)(20,22), (1,5)(3,7)(9,13)(11,15)(18,22)(20,24)>;
 
G:=Group( (1,20,9)(2,21,10)(3,22,11)(4,23,12)(5,24,13)(6,17,14)(7,18,15)(8,19,16), (9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19), (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)(17,21)(18,24)(20,22), (1,5)(3,7)(9,13)(11,15)(18,22)(20,24) );
 
G=PermutationGroup([[(1,20,9),(2,21,10),(3,22,11),(4,23,12),(5,24,13),(6,17,14),(7,18,15),(8,19,16)], [(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19)], [(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),(17,21),(18,24),(20,22)], [(1,5),(3,7),(9,13),(11,15),(18,22),(20,24)]])
 
G:=TransitiveGroup(24,362);
 

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F6A6B6C6D6E8A8B8C8D12A12B12C24A24B
order12222222222234444446666688881212122424
size112334446121212222466122488844121244888

33 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6D6D6C8⋊C22S3×D4S3×D4S3×C8⋊C22
kernelS3×C8⋊C22S3×M4(2)C8⋊D6S3×D8D8⋊S3S3×SD16Q8⋊3D6D12⋊6C22D4⋊D6C3×C8⋊C22C2×S3×D4S3×C4○D4C8⋊C22C4×S3C2×Dic3C22×S3M4(2)D8SD16C2×D4C4○D4S3C4C22C1
# reps1112222111111211122112111

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

-1-1000000
10000000
00-1-10000
00100000
0000-1-100
00001000
000000-1-1
00000010
,
-10000000
11000000
00-100000
00110000
0000-1000
00001100
000000-10
00000011
,
000000-10
0000000-1
0000-1000
00000-100
-10000000
0-1000000
00100000
00010000
,
00100000
00010000
10000000
01000000
0000-1000
00000-100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001

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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