metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊2S3, (C4×S3)⋊3C4, (C4×C12)⋊9C2, C4○(D6⋊C4), C4.22(C4×S3), D6⋊C4.7C2, D6.3(C2×C4), (C2×C4).96D6, C12.25(C2×C4), (C4×Dic3)⋊8C2, C6.3(C4○D4), C6.3(C22×C4), C4○2(Dic3⋊C4), Dic3⋊C4⋊17C2, C3⋊1(C42⋊C2), C2.2(C4○D12), (C2×C6).13C23, Dic3.5(C2×C4), (C2×C12).73C22, C22.10(C22×S3), (C22×S3).14C22, (C2×Dic3).24C22, C2.5(S3×C2×C4), (S3×C2×C4).8C2, (C2×C4)○(D6⋊C4), (C2×C4)○(Dic3⋊C4), SmallGroup(96,79)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊2S3
G = < a,b,c,d | a4=b4=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 154 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊C2, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, S3×C2×C4, C42⋊2S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, S3×C2×C4, C4○D12, C42⋊2S3
►On 48 points
(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 13 29 23)(2 14 30 24)(3 15 31 21)(4 16 32 22)(5 27 43 33)(6 28 44 34)(7 25 41 35)(8 26 42 36)(9 38 48 20)(10 39 45 17)(11 40 46 18)(12 37 47 19)
(1 39 5)(2 40 6)(3 37 7)(4 38 8)(9 36 22)(10 33 23)(11 34 24)(12 35 21)(13 45 27)(14 46 28)(15 47 25)(16 48 26)(17 43 29)(18 44 30)(19 41 31)(20 42 32)
(5 39)(6 40)(7 37)(8 38)(9 34)(10 35)(11 36)(12 33)(13 15)(14 16)(17 43)(18 44)(19 41)(20 42)(21 23)(22 24)(25 45)(26 46)(27 47)(28 48)
G:=sub<Sym(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), (1,13,29,23)(2,14,30,24)(3,15,31,21)(4,16,32,22)(5,27,43,33)(6,28,44,34)(7,25,41,35)(8,26,42,36)(9,38,48,20)(10,39,45,17)(11,40,46,18)(12,37,47,19), (1,39,5)(2,40,6)(3,37,7)(4,38,8)(9,36,22)(10,33,23)(11,34,24)(12,35,21)(13,45,27)(14,46,28)(15,47,25)(16,48,26)(17,43,29)(18,44,30)(19,41,31)(20,42,32), (5,39)(6,40)(7,37)(8,38)(9,34)(10,35)(11,36)(12,33)(13,15)(14,16)(17,43)(18,44)(19,41)(20,42)(21,23)(22,24)(25,45)(26,46)(27,47)(28,48)>;
G:=Group( (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), (1,13,29,23)(2,14,30,24)(3,15,31,21)(4,16,32,22)(5,27,43,33)(6,28,44,34)(7,25,41,35)(8,26,42,36)(9,38,48,20)(10,39,45,17)(11,40,46,18)(12,37,47,19), (1,39,5)(2,40,6)(3,37,7)(4,38,8)(9,36,22)(10,33,23)(11,34,24)(12,35,21)(13,45,27)(14,46,28)(15,47,25)(16,48,26)(17,43,29)(18,44,30)(19,41,31)(20,42,32), (5,39)(6,40)(7,37)(8,38)(9,34)(10,35)(11,36)(12,33)(13,15)(14,16)(17,43)(18,44)(19,41)(20,42)(21,23)(22,24)(25,45)(26,46)(27,47)(28,48) );
G=PermutationGroup([[(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)], [(1,13,29,23),(2,14,30,24),(3,15,31,21),(4,16,32,22),(5,27,43,33),(6,28,44,34),(7,25,41,35),(8,26,42,36),(9,38,48,20),(10,39,45,17),(11,40,46,18),(12,37,47,19)], [(1,39,5),(2,40,6),(3,37,7),(4,38,8),(9,36,22),(10,33,23),(11,34,24),(12,35,21),(13,45,27),(14,46,28),(15,47,25),(16,48,26),(17,43,29),(18,44,30),(19,41,31),(20,42,32)], [(5,39),(6,40),(7,37),(8,38),(9,34),(10,35),(11,36),(12,33),(13,15),(14,16),(17,43),(18,44),(19,41),(20,42),(21,23),(22,24),(25,45),(26,46),(27,47),(28,48)]])
C42⋊2S3 is a maximal subgroup of
D6.C42 C42.243D6 D6.4C42 C42.185D6 C42⋊3D6 C42.30D6 C42.31D6 C4×C4○D12 C42.277D6 S3×C42⋊C2 C42.188D6 C42⋊12D6 C42.93D6 C42.94D6 C42.96D6 C42.97D6 C42.102D6 C42.104D6 C42⋊13D6 C42.108D6 C42⋊14D6 C42⋊18D6 C42.115D6 C42.116D6 C42.122D6 C42.125D6 C42.126D6 C42.232D6 C42.131D6 C42.132D6 C42.133D6 C42.134D6 C42.138D6 C42⋊20D6 C42.141D6 C42⋊23D6 C42.144D6 C42.148D6 C42.151D6 C42.155D6 C42.156D6 C42.160D6 C42⋊26D6 C42.162D6 C42.163D6 C42⋊28D6 C42.168D6 C42.171D6 C42.174D6 C42.176D6 C42.178D6 C42⋊2D9 C62.6C23 C62.25C23 C62.44C23 C62.47C23 C122⋊16C2 (S3×C20)⋊7C4 (C4×D15)⋊10C4 D6.(C4×D5) D30.C2⋊C4 C42⋊2D15 (C4×S3)⋊F5
C42⋊2S3 is a maximal quotient of
C3⋊(C42⋊8C4) C3⋊(C42⋊5C4) C6.(C4×D4) C22.58(S3×D4) D6⋊(C4⋊C4) D6⋊C4⋊5C4 C42.282D6 C42.243D6 C42.182D6 C42.185D6 C4×Dic3⋊C4 C42⋊6Dic3 (C2×C42).6S3 C4×D6⋊C4 (C2×C42)⋊3S3 C42⋊2D9 C62.6C23 C62.25C23 C62.44C23 C62.47C23 C122⋊16C2 (S3×C20)⋊7C4 (C4×D15)⋊10C4 D6.(C4×D5) D30.C2⋊C4 C42⋊2D15 (C4×S3)⋊F5
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | C4○D4 | C4×S3 | C4○D12 |
| kernel | C42⋊2S3 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C4×C12 | S3×C2×C4 | C4×S3 | C42 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 1 | 3 | 4 | 4 | 8 |
Matrix representation of C42⋊2S3 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 8 | 0 | 0 |
| 0 | 2 | 9 |
| 0 | 4 | 11 |
| 1 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 12 |
| 12 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,5,0,0,0,5],[8,0,0,0,2,4,0,9,11],[1,0,0,0,0,1,0,12,12],[12,0,0,0,0,1,0,1,0] >;
C42⋊2S3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2S_3
% in TeX
G:=Group("C4^2:2S3"); // GroupNames label
G:=SmallGroup(96,79);
// by ID
G=gap.SmallGroup(96,79);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,103,362,50,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations