Aliases: C62.(C2xC4), D4.(C32:C4), (D4xC32).C4, C32:7D4.C4, C32:4Q8.C4, C32:7(C8oD4), C62.C4:5C2, C12.D6.4C2, C32:M4(2):6C2, C3:Dic3.32C23, C32:2C8.11C22, C3:S3:3C8:4C2, C4.5(C2xC32:C4), (C3xC12).5(C2xC4), (C2xC32:2C8):8C2, C22.1(C2xC32:C4), C2.8(C22xC32:C4), (C4xC3:S3).37C22, C3:Dic3.23(C2xC4), (C3xC6).30(C22xC4), (C2xC3:Dic3).100C22, (C2xC3:S3).20(C2xC4), SmallGroup(288,935)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.(C2xC4)
G = < a,b,c,d | a6=b6=c2=1, d4=b3, ab=ba, cac=ab3, dad-1=a-1b4, bc=cb, dbd-1=a4b, cd=dc >
Subgroups: 456 in 102 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, D4, D4, Q8, C32, Dic3, C12, D6, C2xC6, C2xC8, M4(2), C4oD4, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C8oD4, C3:Dic3, C3:Dic3, C3xC12, C2xC3:S3, C62, D4:2S3, C32:2C8, C32:2C8, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, C3:S3:3C8, C32:M4(2), C2xC32:2C8, C62.C4, C12.D6, C62.(C2xC4)
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C8oD4, C32:C4, C2xC32:C4, C22xC32:C4, C62.(C2xC4)
Character table of C62.(C2xC4)
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | |
| size | 1 | 1 | 2 | 2 | 18 | 4 | 4 | 2 | 9 | 9 | 18 | 18 | 4 | 4 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 8 | 8 | |
| 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 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | i | -i | -i | -i | -i | i | -i | i | i | 1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | -i | i | i | i | -i | i | -i | i | -i | 1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -i | -i | i | i | -i | -i | i | i | -i | i | -1 | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | i | -i | i | i | -i | -i | -1 | -1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | -i | i | i | i | i | -i | i | -i | -i | 1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | i | -i | -i | -i | i | -i | i | -i | i | 1 | 1 | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | i | i | -i | -i | i | i | -i | -i | i | -i | -1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | -i | i | -i | -i | i | i | -1 | -1 | linear of order 4 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ87 | 2ζ83 | 2ζ8 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ85 | 2ζ8 | 2ζ83 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ83 | 2ζ87 | 2ζ85 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2ζ8 | 2ζ85 | 2ζ87 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8oD4 |
| ρ21 | 4 | 4 | 4 | 4 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | orthogonal lifted from C32:C4 |
| ρ22 | 4 | 4 | -4 | 4 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | -1 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | orthogonal lifted from C2xC32:C4 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | orthogonal lifted from C32:C4 |
| ρ24 | 4 | 4 | -4 | 4 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | 1 | 2 | -1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | orthogonal lifted from C2xC32:C4 |
| ρ25 | 4 | 4 | 4 | -4 | 0 | 1 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | -2 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | orthogonal lifted from C2xC32:C4 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 1 | -2 | 4 | 0 | 0 | 0 | 0 | -2 | 1 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | orthogonal lifted from C2xC32:C4 |
| ρ27 | 4 | 4 | 4 | -4 | 0 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 1 | -2 | 2 | 1 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | orthogonal lifted from C2xC32:C4 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 1 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | orthogonal lifted from C2xC32:C4 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 8 | -8 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 39 17 31 13 48)(2 41 14 32 18 40)(3 42 15 25 19 33)(4 34 20 26 16 43)(5 35 21 27 9 44)(6 45 10 28 22 36)(7 46 11 29 23 37)(8 38 24 30 12 47)
(1 5)(2 10 18 6 14 22)(3 7)(4 24 16 8 20 12)(9 13)(11 15)(17 21)(19 23)(25 29)(26 47 34 30 43 38)(27 31)(28 40 45 32 36 41)(33 37)(35 39)(42 46)(44 48)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)
(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)
G:=sub<Sym(48)| (1,39,17,31,13,48)(2,41,14,32,18,40)(3,42,15,25,19,33)(4,34,20,26,16,43)(5,35,21,27,9,44)(6,45,10,28,22,36)(7,46,11,29,23,37)(8,38,24,30,12,47), (1,5)(2,10,18,6,14,22)(3,7)(4,24,16,8,20,12)(9,13)(11,15)(17,21)(19,23)(25,29)(26,47,34,30,43,38)(27,31)(28,40,45,32,36,41)(33,37)(35,39)(42,46)(44,48), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40), (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)>;
G:=Group( (1,39,17,31,13,48)(2,41,14,32,18,40)(3,42,15,25,19,33)(4,34,20,26,16,43)(5,35,21,27,9,44)(6,45,10,28,22,36)(7,46,11,29,23,37)(8,38,24,30,12,47), (1,5)(2,10,18,6,14,22)(3,7)(4,24,16,8,20,12)(9,13)(11,15)(17,21)(19,23)(25,29)(26,47,34,30,43,38)(27,31)(28,40,45,32,36,41)(33,37)(35,39)(42,46)(44,48), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40), (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) );
G=PermutationGroup([[(1,39,17,31,13,48),(2,41,14,32,18,40),(3,42,15,25,19,33),(4,34,20,26,16,43),(5,35,21,27,9,44),(6,45,10,28,22,36),(7,46,11,29,23,37),(8,38,24,30,12,47)], [(1,5),(2,10,18,6,14,22),(3,7),(4,24,16,8,20,12),(9,13),(11,15),(17,21),(19,23),(25,29),(26,47,34,30,43,38),(27,31),(28,40,45,32,36,41),(33,37),(35,39),(42,46),(44,48)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40)], [(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)]])
Matrix representation of C62.(C2xC4) ►in GL6(F73)
| 0 | 46 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 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 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(73))| [0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C62.(C2xC4) in GAP, Magma, Sage, TeX
C_6^2.(C_2\times C_4)
% in TeX
G:=Group("C6^2.(C2xC4)"); // GroupNames label
G:=SmallGroup(288,935);
// by ID
G=gap.SmallGroup(288,935);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,219,80,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^2=1,d^4=b^3,a*b=b*a,c*a*c=a*b^3,d*a*d^-1=a^-1*b^4,b*c=c*b,d*b*d^-1=a^4*b,c*d=d*c>;
// generators/relations
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