direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4⋊C4, D6.Q8, D6.11D4, C4⋊3(C4×S3), (C4×S3)⋊1C4, C12⋊1(C2×C4), C2.3(S3×D4), C2.2(S3×Q8), D6.8(C2×C4), C6.23(C2×D4), (C2×C4).30D6, C6.12(C2×Q8), C4⋊Dic3⋊11C2, Dic3⋊3(C2×C4), C6.9(C22×C4), Dic3⋊C4⋊11C2, (C2×C6).32C23, (C2×C12).23C22, C22.16(C22×S3), (C22×S3).34C22, (C2×Dic3).29C22, C3⋊1(C2×C4⋊C4), (C3×C4⋊C4)⋊2C2, (S3×C2×C4).1C2, C2.11(S3×C2×C4), SmallGroup(96,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4⋊C4
G = < a,b,c,d | a3=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 186 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C4⋊C4, C4⋊C4, C22×C4, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, S3×C4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, S3×C4⋊C4
Character table of S3×C4⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | -i | i | i | -i | 1 | -i | -i | 1 | -1 | i | i | -1 | -1 | 1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | -1 | i | -i | 1 | -1 | -i | i | -1 | -1 | 1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | i | -i | 1 | i | i | -1 | 1 | -i | -i | -1 | -1 | 1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -i | i | -1 | 1 | i | -i | -1 | -1 | 1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -i | -i | i | 1 | i | i | 1 | -1 | -i | -i | -1 | -1 | 1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | -1 | -i | i | 1 | -1 | i | -i | -1 | -1 | 1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | -i | i | 1 | -i | -i | -1 | 1 | i | i | -1 | -1 | 1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -1 | i | -i | -1 | 1 | -i | i | -1 | -1 | 1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ22 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ23 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 2i | -2i | 2i | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | i | 1 | -i | -1 | complex lifted from C4×S3 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2i | 2i | -2i | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | -i | 1 | i | -1 | complex lifted from C4×S3 |
| ρ27 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | -2i | 2i | 2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | -i | -i | -1 | i | 1 | complex lifted from C4×S3 |
| ρ28 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 2i | -2i | -2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | i | i | -1 | -i | 1 | complex lifted from C4×S3 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
►On 48 points
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 29 34)(6 30 35)(7 31 36)(8 32 33)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)
(1 3)(2 4)(5 7)(6 8)(9 27)(10 28)(11 25)(12 26)(13 20)(14 17)(15 18)(16 19)(21 39)(22 40)(23 37)(24 38)(29 36)(30 33)(31 34)(32 35)(41 43)(42 44)(45 47)(46 48)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 47 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 29 15 23)(10 32 16 22)(11 31 13 21)(12 30 14 24)(17 38 26 33)(18 37 27 36)(19 40 28 35)(20 39 25 34)
G:=sub<Sym(48)| (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (1,3)(2,4)(5,7)(6,8)(9,27)(10,28)(11,25)(12,26)(13,20)(14,17)(15,18)(16,19)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(41,43)(42,44)(45,47)(46,48), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34)>;
G:=Group( (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,29,34)(6,30,35)(7,31,36)(8,32,33)(9,25,42)(10,26,43)(11,27,44)(12,28,41)(21,37,46)(22,38,47)(23,39,48)(24,40,45), (1,3)(2,4)(5,7)(6,8)(9,27)(10,28)(11,25)(12,26)(13,20)(14,17)(15,18)(16,19)(21,39)(22,40)(23,37)(24,38)(29,36)(30,33)(31,34)(32,35)(41,43)(42,44)(45,47)(46,48), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,47,43,8)(2,46,44,7)(3,45,41,6)(4,48,42,5)(9,29,15,23)(10,32,16,22)(11,31,13,21)(12,30,14,24)(17,38,26,33)(18,37,27,36)(19,40,28,35)(20,39,25,34) );
G=PermutationGroup([[(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,29,34),(6,30,35),(7,31,36),(8,32,33),(9,25,42),(10,26,43),(11,27,44),(12,28,41),(21,37,46),(22,38,47),(23,39,48),(24,40,45)], [(1,3),(2,4),(5,7),(6,8),(9,27),(10,28),(11,25),(12,26),(13,20),(14,17),(15,18),(16,19),(21,39),(22,40),(23,37),(24,38),(29,36),(30,33),(31,34),(32,35),(41,43),(42,44),(45,47),(46,48)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,47,43,8),(2,46,44,7),(3,45,41,6),(4,48,42,5),(9,29,15,23),(10,32,16,22),(11,31,13,21),(12,30,14,24),(17,38,26,33),(18,37,27,36),(19,40,28,35),(20,39,25,34)]])
S3×C4⋊C4 is a maximal subgroup of
D4⋊(C4×S3) D6.D8 D6.SD16 Q8⋊7(C4×S3) D6.1SD16 D6.Q16 C8⋊(C4×S3) D6.2SD16 D6.4SD16 C8⋊S3⋊C4 D6.5D8 D6.2Q16 C6.82+ 1+4 C6.2- 1+4 C6.102+ 1+4 C42.91D6 C42.94D6 C42.95D6 C4×S3×D4 C42.108D6 D12⋊24D4 C42.113D6 C4×S3×Q8 C42.126D6 D12⋊10Q8 C42.132D6 C6.722- 1+4 C6.732- 1+4 C6.432+ 1+4 C6.172- 1+4 D12⋊22D4 C6.1182+ 1+4 C6.522+ 1+4 C6.202- 1+4 C6.212- 1+4 C6.822- 1+4 C6.632+ 1+4 C6.642+ 1+4 C42.148D6 D12⋊7Q8 C42.150D6 C42.151D6 C42.152D6 C42.153D6 C42.161D6 C42.162D6 C42.163D6 D12⋊12D4 C42.174D6 D12⋊9Q8 C62.53C23 C62.70C23 D30.Q8 D30.2Q8
S3×C4⋊C4 is a maximal quotient of
C6.(C4×Q8) Dic3⋊C42 C3⋊(C42⋊8C4) D6⋊(C4⋊C4) D6⋊C4⋊C4 C12⋊M4(2) C42.30D6 (S3×C8)⋊C4 C8⋊(C4×S3) C8.27(C4×S3) C8⋊S3⋊C4 M4(2).25D6 (C4×Dic3)⋊8C4 Dic3⋊(C4⋊C4) C4⋊(D6⋊C4) D6⋊C4⋊6C4 C62.53C23 C62.70C23 D30.Q8 D30.2Q8
Matrix representation of S3×C4⋊C4 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 9 | 3 | 0 | 0 |
| 3 | 4 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,12],[12,0,0,0,0,12,0,0,0,0,0,1,0,0,1,0],[9,3,0,0,3,4,0,0,0,0,12,0,0,0,0,12],[0,12,0,0,1,0,0,0,0,0,8,0,0,0,0,8] >;
S3×C4⋊C4 in GAP, Magma, Sage, TeX
S_3\times C_4\rtimes C_4
% in TeX
G:=Group("S3xC4:C4"); // GroupNames label
G:=SmallGroup(96,98);
// by ID
G=gap.SmallGroup(96,98);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,188,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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