direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C23⋊C4, (C2×D12)⋊4C4, C23⋊2(C4×S3), (S3×C23)⋊3C4, C22⋊C4⋊22D6, (C2×D4).120D6, C22.26(S3×D4), (C22×S3).10D4, (C6×D4).10C22, C23.7D6⋊4C2, C23.6D6⋊4C2, (C22×C6).4C23, D6.15(C22⋊C4), C6.D4⋊3C22, C23.14(C22×S3), (S3×C23).10C22, (C2×C12)⋊(C2×C4), (S3×C2×C4)⋊2C4, (C2×C4)⋊1(C4×S3), (C22×C6)⋊(C2×C4), C3⋊1(C2×C23⋊C4), (C2×S3×D4).1C2, (C2×C3⋊D4)⋊2C4, (C3×C23⋊C4)⋊6C2, (C2×C6).19(C2×D4), C22.13(S3×C2×C4), (S3×C22⋊C4)⋊22C2, (C22×S3)⋊1(C2×C4), (C2×Dic3)⋊1(C2×C4), C6.11(C2×C22⋊C4), C2.12(S3×C22⋊C4), (C2×C6).7(C22×C4), (C2×C3⋊D4).4C22, (C3×C22⋊C4)⋊33C22, SmallGroup(192,302)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C23⋊C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >
Subgroups: 768 in 210 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C22×C4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C23⋊C4, C23⋊C4, C2×C22⋊C4, C22×D4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C6×D4, S3×C23, C2×C23⋊C4, C23.6D6, C23.7D6, C3×C23⋊C4, S3×C22⋊C4, C2×S3×D4, S3×C23⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C23⋊C4, C2×C22⋊C4, S3×C2×C4, S3×D4, C2×C23⋊C4, S3×C22⋊C4, S3×C23⋊C4
►On 24 points - transitive group 24T337
Generators in S24
(1 4 11)(2 3 12)(5 10 8)(6 9 7)(13 23 17)(14 24 18)(15 21 19)(16 22 20)
(1 10)(2 9)(3 6)(4 5)(7 12)(8 11)(13 19)(14 20)(15 17)(16 18)(21 23)(22 24)
(1 19)(2 18)(3 14)(4 15)(5 17)(6 20)(7 22)(8 23)(9 16)(10 13)(11 21)(12 24)
(2 6)(3 9)(7 12)(14 16)(18 20)(22 24)
(1 5)(2 6)(3 9)(4 10)(7 12)(8 11)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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)
G:=sub<Sym(24)| (1,4,11)(2,3,12)(5,10,8)(6,9,7)(13,23,17)(14,24,18)(15,21,19)(16,22,20), (1,10)(2,9)(3,6)(4,5)(7,12)(8,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24), (1,19)(2,18)(3,14)(4,15)(5,17)(6,20)(7,22)(8,23)(9,16)(10,13)(11,21)(12,24), (2,6)(3,9)(7,12)(14,16)(18,20)(22,24), (1,5)(2,6)(3,9)(4,10)(7,12)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;
G:=Group( (1,4,11)(2,3,12)(5,10,8)(6,9,7)(13,23,17)(14,24,18)(15,21,19)(16,22,20), (1,10)(2,9)(3,6)(4,5)(7,12)(8,11)(13,19)(14,20)(15,17)(16,18)(21,23)(22,24), (1,19)(2,18)(3,14)(4,15)(5,17)(6,20)(7,22)(8,23)(9,16)(10,13)(11,21)(12,24), (2,6)(3,9)(7,12)(14,16)(18,20)(22,24), (1,5)(2,6)(3,9)(4,10)(7,12)(8,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );
G=PermutationGroup([[(1,4,11),(2,3,12),(5,10,8),(6,9,7),(13,23,17),(14,24,18),(15,21,19),(16,22,20)], [(1,10),(2,9),(3,6),(4,5),(7,12),(8,11),(13,19),(14,20),(15,17),(16,18),(21,23),(22,24)], [(1,19),(2,18),(3,14),(4,15),(5,17),(6,20),(7,22),(8,23),(9,16),(10,13),(11,21),(12,24)], [(2,6),(3,9),(7,12),(14,16),(18,20),(22,24)], [(1,5),(2,6),(3,9),(4,10),(7,12),(8,11),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)]])
G:=TransitiveGroup(24,337);
►On 24 points - transitive group 24T341
Generators in S24
(1 16 9)(2 13 10)(3 14 11)(4 15 12)(5 24 19)(6 21 20)(7 22 17)(8 23 18)
(1 8)(2 5)(3 6)(4 7)(9 23)(10 24)(11 21)(12 22)(13 19)(14 20)(15 17)(16 18)
(2 5)(3 6)(10 19)(11 20)(13 24)(14 21)
(2 5)(4 7)(10 19)(12 17)(13 24)(15 22)
(1 8)(2 5)(3 6)(4 7)(9 18)(10 19)(11 20)(12 17)(13 24)(14 21)(15 22)(16 23)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
G:=sub<Sym(24)| (1,16,9)(2,13,10)(3,14,11)(4,15,12)(5,24,19)(6,21,20)(7,22,17)(8,23,18), (1,8)(2,5)(3,6)(4,7)(9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,17)(16,18), (2,5)(3,6)(10,19)(11,20)(13,24)(14,21), (2,5)(4,7)(10,19)(12,17)(13,24)(15,22), (1,8)(2,5)(3,6)(4,7)(9,18)(10,19)(11,20)(12,17)(13,24)(14,21)(15,22)(16,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;
G:=Group( (1,16,9)(2,13,10)(3,14,11)(4,15,12)(5,24,19)(6,21,20)(7,22,17)(8,23,18), (1,8)(2,5)(3,6)(4,7)(9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,17)(16,18), (2,5)(3,6)(10,19)(11,20)(13,24)(14,21), (2,5)(4,7)(10,19)(12,17)(13,24)(15,22), (1,8)(2,5)(3,6)(4,7)(9,18)(10,19)(11,20)(12,17)(13,24)(14,21)(15,22)(16,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );
G=PermutationGroup([[(1,16,9),(2,13,10),(3,14,11),(4,15,12),(5,24,19),(6,21,20),(7,22,17),(8,23,18)], [(1,8),(2,5),(3,6),(4,7),(9,23),(10,24),(11,21),(12,22),(13,19),(14,20),(15,17),(16,18)], [(2,5),(3,6),(10,19),(11,20),(13,24),(14,21)], [(2,5),(4,7),(10,19),(12,17),(13,24),(15,22)], [(1,8),(2,5),(3,6),(4,7),(9,18),(10,19),(11,20),(12,17),(13,24),(14,21),(15,22),(16,23)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)]])
G:=TransitiveGroup(24,341);
►On 24 points - transitive group 24T363
Generators in S24
(1 19 21)(2 20 22)(3 17 23)(4 18 24)(5 14 9)(6 15 10)(7 16 11)(8 13 12)
(1 14)(2 15)(3 16)(4 13)(5 19)(6 20)(7 17)(8 18)(9 21)(10 22)(11 23)(12 24)
(1 4)(2 16)(3 15)(5 8)(6 23)(7 22)(9 12)(10 17)(11 20)(13 14)(18 19)(21 24)
(1 16)(2 4)(3 14)(5 23)(6 8)(7 21)(9 17)(10 12)(11 19)(13 15)(18 20)(22 24)
(1 14)(2 15)(3 16)(4 13)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
G:=sub<Sym(24)| (1,19,21)(2,20,22)(3,17,23)(4,18,24)(5,14,9)(6,15,10)(7,16,11)(8,13,12), (1,14)(2,15)(3,16)(4,13)(5,19)(6,20)(7,17)(8,18)(9,21)(10,22)(11,23)(12,24), (1,4)(2,16)(3,15)(5,8)(6,23)(7,22)(9,12)(10,17)(11,20)(13,14)(18,19)(21,24), (1,16)(2,4)(3,14)(5,23)(6,8)(7,21)(9,17)(10,12)(11,19)(13,15)(18,20)(22,24), (1,14)(2,15)(3,16)(4,13)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;
G:=Group( (1,19,21)(2,20,22)(3,17,23)(4,18,24)(5,14,9)(6,15,10)(7,16,11)(8,13,12), (1,14)(2,15)(3,16)(4,13)(5,19)(6,20)(7,17)(8,18)(9,21)(10,22)(11,23)(12,24), (1,4)(2,16)(3,15)(5,8)(6,23)(7,22)(9,12)(10,17)(11,20)(13,14)(18,19)(21,24), (1,16)(2,4)(3,14)(5,23)(6,8)(7,21)(9,17)(10,12)(11,19)(13,15)(18,20)(22,24), (1,14)(2,15)(3,16)(4,13)(5,21)(6,22)(7,23)(8,24)(9,19)(10,20)(11,17)(12,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );
G=PermutationGroup([[(1,19,21),(2,20,22),(3,17,23),(4,18,24),(5,14,9),(6,15,10),(7,16,11),(8,13,12)], [(1,14),(2,15),(3,16),(4,13),(5,19),(6,20),(7,17),(8,18),(9,21),(10,22),(11,23),(12,24)], [(1,4),(2,16),(3,15),(5,8),(6,23),(7,22),(9,12),(10,17),(11,20),(13,14),(18,19),(21,24)], [(1,16),(2,4),(3,14),(5,23),(6,8),(7,21),(9,17),(10,12),(11,19),(13,15),(18,20),(22,24)], [(1,14),(2,15),(3,16),(4,13),(5,21),(6,22),(7,23),(8,24),(9,19),(10,20),(11,17),(12,18)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)]])
G:=TransitiveGroup(24,363);
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4E | 4F | ··· | 4J | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12E |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 4 | 4 | 4 | 8 | 8 | ··· | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | C4×S3 | C23⋊C4 | S3×D4 | S3×C23⋊C4 |
| kernel | S3×C23⋊C4 | C23.6D6 | C23.7D6 | C3×C23⋊C4 | S3×C22⋊C4 | C2×S3×D4 | S3×C2×C4 | C2×D12 | C2×C3⋊D4 | S3×C23 | C23⋊C4 | C22×S3 | C22⋊C4 | C2×D4 | C2×C4 | C23 | S3 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 4 | 2 | 1 | 2 | 2 | 2 | 2 | 1 |
Matrix representation of S3×C23⋊C4 ►in GL6(𝔽13)
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 12 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 12 | 8 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 0 | 10 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 12 | 8 |
| 0 | 0 | 0 | 0 | 3 | 1 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1,12,0,0,0,0,0,8,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,3,0,0,0,12,0,0,0,0,0,0,1,10,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,1,0,0,0,1,0,1,0,0,0,0,0,12,3,0,0,0,0,8,1] >;
S3×C23⋊C4 in GAP, Magma, Sage, TeX
S_3\times C_2^3\rtimes C_4
% in TeX
G:=Group("S3xC2^3:C4"); // GroupNames label
G:=SmallGroup(192,302);
// by ID
G=gap.SmallGroup(192,302);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,58,570,438,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations