non-abelian, soluble, monomial
Aliases: C62.1Q8, C22.2PSU3(𝔽2), C3⋊S3.3C42, C32⋊2(C2.C42), C2.2(C2.PSU3(𝔽2)), (C2×C32⋊C4)⋊2C4, (C3×C6).9(C4⋊C4), (C2×C3⋊S3).12D4, C3⋊S3.5(C22⋊C4), (C22×C32⋊C4).2C2, (C22×C3⋊S3).3C22, (C2×C3⋊S3).12(C2×C4), SmallGroup(288,395)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.Q8
G = < a,b,c,d | a6=b6=c4=1, d2=a3c2, ab=ba, cac-1=a3b2, dad-1=a-1b4, cbc-1=a4b3, dbd-1=a4b, dcd-1=a3b3c-1 >
Subgroups: 612 in 92 conjugacy classes, 31 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C23, C32, D6, C2×C6, C22×C4, C3⋊S3, C3⋊S3, C3×C6, C22×S3, C2.C42, C32⋊C4, C2×C3⋊S3, C62, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C22×C32⋊C4, C62.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, PSU3(𝔽2), C2.PSU3(𝔽2), C62.Q8
Character table of C62.Q8
| 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 | |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 8 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 8 | 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 | 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 | i | i | -i | -i | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | -1 | -1 | 1 | 1 | i | i | -i | -i | 1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | 1 | 1 | -1 | i | -i | -i | i | -i | i | i | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | -1 | -1 | 1 | i | -i | -i | i | i | -i | -i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | -1 | -1 | 1 | -i | i | i | -i | -i | i | i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ10 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | i | i | 1 | 1 | -1 | -1 | -i | -i | i | -i | 1 | -1 | -1 | linear of order 4 |
| ρ11 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | -i | 1 | 1 | -1 | -1 | i | i | -i | i | 1 | -1 | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | 1 | 1 | -1 | -i | i | i | -i | i | -i | -i | -1 | -1 | 1 | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | i | -i | i | -i | i | -1 | 1 | -1 | -i | -1 | -1 | 1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | -i | -i | i | -i | i | 1 | -1 | 1 | i | -1 | -1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -i | i | -i | -1 | 1 | -1 | i | -1 | -1 | 1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -i | i | i | -i | i | -i | 1 | -1 | 1 | -i | -1 | -1 | 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 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | orthogonal lifted from D4 |
| 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 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 8 | 8 | -8 | -8 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | orthogonal lifted from C2.PSU3(𝔽2) |
| ρ22 | 8 | -8 | 8 | -8 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from C2.PSU3(𝔽2) |
| ρ23 | 8 | -8 | -8 | 8 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2.PSU3(𝔽2) |
| ρ24 | 8 | 8 | 8 | 8 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from PSU3(𝔽2) |
►On 48 points
Generators in S48
(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 12 7 9 3 6)(2 11 8 10 4 5)(13 43 17 47 15 45)(14 44 18 48 16 46)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 38 33 40 35 42)(32 39 34 41 36 37)
(1 27 10 22)(2 30 9 19)(3 29 5 20)(4 26 6 23)(7 25 11 24)(8 28 12 21)(13 38 46 34)(14 39 45 33)(15 40 44 32)(16 41 43 31)(17 42 48 36)(18 37 47 35)
(1 39 9 36)(2 42 10 33)(3 37 6 32)(4 40 5 35)(7 41 12 34)(8 38 11 31)(13 29 43 23)(14 30 48 22)(15 25 47 21)(16 26 46 20)(17 27 45 19)(18 28 44 24)
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), (1,12,7,9,3,6)(2,11,8,10,4,5)(13,43,17,47,15,45)(14,44,18,48,16,46)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,38,33,40,35,42)(32,39,34,41,36,37), (1,27,10,22)(2,30,9,19)(3,29,5,20)(4,26,6,23)(7,25,11,24)(8,28,12,21)(13,38,46,34)(14,39,45,33)(15,40,44,32)(16,41,43,31)(17,42,48,36)(18,37,47,35), (1,39,9,36)(2,42,10,33)(3,37,6,32)(4,40,5,35)(7,41,12,34)(8,38,11,31)(13,29,43,23)(14,30,48,22)(15,25,47,21)(16,26,46,20)(17,27,45,19)(18,28,44,24)>;
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), (1,12,7,9,3,6)(2,11,8,10,4,5)(13,43,17,47,15,45)(14,44,18,48,16,46)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,38,33,40,35,42)(32,39,34,41,36,37), (1,27,10,22)(2,30,9,19)(3,29,5,20)(4,26,6,23)(7,25,11,24)(8,28,12,21)(13,38,46,34)(14,39,45,33)(15,40,44,32)(16,41,43,31)(17,42,48,36)(18,37,47,35), (1,39,9,36)(2,42,10,33)(3,37,6,32)(4,40,5,35)(7,41,12,34)(8,38,11,31)(13,29,43,23)(14,30,48,22)(15,25,47,21)(16,26,46,20)(17,27,45,19)(18,28,44,24) );
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)], [(1,12,7,9,3,6),(2,11,8,10,4,5),(13,43,17,47,15,45),(14,44,18,48,16,46),(19,27),(20,28),(21,29),(22,30),(23,25),(24,26),(31,38,33,40,35,42),(32,39,34,41,36,37)], [(1,27,10,22),(2,30,9,19),(3,29,5,20),(4,26,6,23),(7,25,11,24),(8,28,12,21),(13,38,46,34),(14,39,45,33),(15,40,44,32),(16,41,43,31),(17,42,48,36),(18,37,47,35)], [(1,39,9,36),(2,42,10,33),(3,37,6,32),(4,40,5,35),(7,41,12,34),(8,38,11,31),(13,29,43,23),(14,30,48,22),(15,25,47,21),(16,26,46,20),(17,27,45,19),(18,28,44,24)]])
Matrix representation of C62.Q8 ►in GL10(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 1 | 0 | 0 | 5 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 8 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 5 | 0 | 0 | 12 |
| 0 | 0 | 7 | 0 | 12 | 12 | 0 | 8 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 1 | 1 | 5 | 5 | 11 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 1 | 1 | 8 | 8 | 0 | 0 | 8 | 0 |
| 0 | 0 | 1 | 1 | 8 | 8 | 12 | 0 | 8 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 6 | 6 | 1 | 1 | 5 | 5 | 11 | 12 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 5 | 5 | 8 | 8 | 12 | 0 |
| 0 | 0 | 9 | 9 | 5 | 4 | 8 | 8 | 12 | 0 |
G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,12,0,0,0,12,0,0,0,0,0,0,12,12,0,8,0,0,0,0,0,0,1,0,5,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,7,0,0,1,1,0,0,0,0,6,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,1,1,5,0,0,0,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,6,0,1,1,0,0,0,0,12,0,6,0,1,1,0,0,1,0,0,0,1,0,8,8,0,0,0,1,0,0,1,0,8,8,0,0,0,0,0,0,5,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,11,12,8,8,0,0,0,0,0,0,12,1,0,0],[8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,12,9,9,0,0,0,0,0,6,12,0,9,9,0,0,0,0,0,1,0,0,5,5,0,0,0,0,0,1,0,0,5,4,0,0,1,0,0,5,0,0,8,8,0,0,0,1,0,5,0,0,8,8,0,0,0,0,12,11,0,0,12,12,0,0,0,0,1,12,0,0,0,0] >;
C62.Q8 in GAP, Magma, Sage, TeX
C_6^2.Q_8
% in TeX
G:=Group("C6^2.Q8"); // GroupNames label
G:=SmallGroup(288,395);
// by ID
G=gap.SmallGroup(288,395);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,92,9413,2028,691,12550,1581,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=a^3*c^2,a*b=b*a,c*a*c^-1=a^3*b^2,d*a*d^-1=a^-1*b^4,c*b*c^-1=a^4*b^3,d*b*d^-1=a^4*b,d*c*d^-1=a^3*b^3*c^-1>;
// generators/relations
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