metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4:C4:6S3, D6:C4.5C2, C4:Dic3:7C2, (C2xC4).32D6, Dic3:C4:13C2, C3:3(C42:2C2), (C4xDic3):14C2, C6.14(C4oD4), (C2xC12).7C22, (C2xC6).39C23, C2.16(C4oD12), C2.7(Q8:3S3), C2.14(D4:2S3), (C22xS3).8C22, C22.53(C22xS3), (C2xDic3).32C22, (C3xC4:C4):9C2, SmallGroup(96,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4:C4:S3
G = < a,b,c,d | a4=b4=c3=d2=1, bab-1=a-1, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 138 in 60 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2xC4, C2xC4, C23, Dic3, C12, D6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xDic3, C2xC12, C22xS3, C42:2C2, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C3xC4:C4, C4:C4:S3
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, C42:2C2, C4oD12, D4:2S3, Q8:3S3, C4:C4:S3
Character table of C4:C4:S3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | complex lifted from C4oD4 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | complex lifted from C4oD4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | -i | √3 | -√3 | i | √-3 | complex lifted from C4oD12 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-3 | i | -√3 | √3 | -i | √-3 | complex lifted from C4oD12 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | -i | -√3 | √3 | i | -√-3 | complex lifted from C4oD12 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-3 | i | √3 | -√3 | -i | -√-3 | complex lifted from C4oD12 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8:3S3, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4:2S3, Schur index 2 |
►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 45 5 42)(2 48 6 41)(3 47 7 44)(4 46 8 43)(9 19 34 31)(10 18 35 30)(11 17 36 29)(12 20 33 32)(13 24 38 26)(14 23 39 25)(15 22 40 28)(16 21 37 27)
(1 35 39)(2 36 40)(3 33 37)(4 34 38)(5 10 14)(6 11 15)(7 12 16)(8 9 13)(17 22 41)(18 23 42)(19 24 43)(20 21 44)(25 45 30)(26 46 31)(27 47 32)(28 48 29)
(2 6)(4 8)(9 38)(10 14)(11 40)(12 16)(13 34)(15 36)(17 26)(18 21)(19 28)(20 23)(22 31)(24 29)(25 32)(27 30)(33 37)(35 39)(41 46)(42 44)(43 48)(45 47)
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), (1,45,5,42)(2,48,6,41)(3,47,7,44)(4,46,8,43)(9,19,34,31)(10,18,35,30)(11,17,36,29)(12,20,33,32)(13,24,38,26)(14,23,39,25)(15,22,40,28)(16,21,37,27), (1,35,39)(2,36,40)(3,33,37)(4,34,38)(5,10,14)(6,11,15)(7,12,16)(8,9,13)(17,22,41)(18,23,42)(19,24,43)(20,21,44)(25,45,30)(26,46,31)(27,47,32)(28,48,29), (2,6)(4,8)(9,38)(10,14)(11,40)(12,16)(13,34)(15,36)(17,26)(18,21)(19,28)(20,23)(22,31)(24,29)(25,32)(27,30)(33,37)(35,39)(41,46)(42,44)(43,48)(45,47)>;
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), (1,45,5,42)(2,48,6,41)(3,47,7,44)(4,46,8,43)(9,19,34,31)(10,18,35,30)(11,17,36,29)(12,20,33,32)(13,24,38,26)(14,23,39,25)(15,22,40,28)(16,21,37,27), (1,35,39)(2,36,40)(3,33,37)(4,34,38)(5,10,14)(6,11,15)(7,12,16)(8,9,13)(17,22,41)(18,23,42)(19,24,43)(20,21,44)(25,45,30)(26,46,31)(27,47,32)(28,48,29), (2,6)(4,8)(9,38)(10,14)(11,40)(12,16)(13,34)(15,36)(17,26)(18,21)(19,28)(20,23)(22,31)(24,29)(25,32)(27,30)(33,37)(35,39)(41,46)(42,44)(43,48)(45,47) );
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)], [(1,45,5,42),(2,48,6,41),(3,47,7,44),(4,46,8,43),(9,19,34,31),(10,18,35,30),(11,17,36,29),(12,20,33,32),(13,24,38,26),(14,23,39,25),(15,22,40,28),(16,21,37,27)], [(1,35,39),(2,36,40),(3,33,37),(4,34,38),(5,10,14),(6,11,15),(7,12,16),(8,9,13),(17,22,41),(18,23,42),(19,24,43),(20,21,44),(25,45,30),(26,46,31),(27,47,32),(28,48,29)], [(2,6),(4,8),(9,38),(10,14),(11,40),(12,16),(13,34),(15,36),(17,26),(18,21),(19,28),(20,23),(22,31),(24,29),(25,32),(27,30),(33,37),(35,39),(41,46),(42,44),(43,48),(45,47)]])
C4:C4:S3 is a maximal subgroup of
C6.52- 1+4 C6.112+ 1+4 C6.62- 1+4 C42:12D6 C42.93D6 C42.96D6 C42.99D6 C42.102D6 C42:18D6 C42:19D6 C42.119D6 C42.122D6 C42.131D6 C42.134D6 C42.135D6 C6.342+ 1+4 C6.422+ 1+4 C6.462+ 1+4 C6.482+ 1+4 C4:C4.187D6 C6.532+ 1+4 C6.202- 1+4 C6.222- 1+4 C6.232- 1+4 C6.772- 1+4 C6.242- 1+4 C6.562+ 1+4 C6.782- 1+4 C6.252- 1+4 C4:C4.197D6 C6.612+ 1+4 C6.1222+ 1+4 C6.642+ 1+4 C6.652+ 1+4 C6.852- 1+4 C6.682+ 1+4 C42.237D6 C42.150D6 C42.151D6 C42.152D6 C42.153D6 C42.154D6 C42.156D6 C42.157D6 C42.160D6 S3xC42:2C2 C42:25D6 C42.161D6 C42.163D6 C42.164D6 C42.165D6 C42.176D6 C42.177D6 C42.178D6 C42.180D6 C4:C4:D9 C62.18C23 C62.28C23 C62.31C23 C62.32C23 C62.38C23 C62.242C23 C4:Dic3:D5 D6:C4.D5 C60:5C4:C2 C4:Dic5:S3 D6:Dic5.C2 C10.D4:S3 C4:C4:D15
C4:C4:S3 is a maximal quotient of
C3:(C42:5C4) C2.(C4xDic6) (C2xC4).Dic6 (C22xC4).30D6 D6:C4:5C4 D6:C4:3C4 (C2xC4).21D12 C6.(C4:D4) C6.67(C4xD4) C4:C4:5Dic3 (C2xC12).288D4 (C2xC12).55D4 D6:C4:7C4 (C2xC12).289D4 (C2xC12).56D4 C4:C4:D9 C62.18C23 C62.28C23 C62.31C23 C62.32C23 C62.38C23 C62.242C23 C4:Dic3:D5 D6:C4.D5 C60:5C4:C2 C4:Dic5:S3 D6:Dic5.C2 C10.D4:S3 C4:C4:D15
Matrix representation of C4:C4:S3 ►in GL4(F13) generated by
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 7 | 10 |
| 0 | 0 | 8 | 6 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 11 | 9 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 9 | 12 |
G:=sub<GL(4,GF(13))| [11,4,0,0,9,2,0,0,0,0,7,8,0,0,10,6],[8,0,0,0,0,8,0,0,0,0,4,11,0,0,2,9],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,1,9,0,0,0,12] >;
C4:C4:S3 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes S_3
% in TeX
G:=Group("C4:C4:S3"); // GroupNames label
G:=SmallGroup(96,105);
// by ID
G=gap.SmallGroup(96,105);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,55,218,188,86,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^3=d^2=1,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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