non-abelian, soluble, monomial
Aliases: C62.13D4, C32⋊D8⋊2C2, C32⋊Q16⋊3C2, C22.1S3≀C2, C32⋊2(C4○D8), D6.4D6⋊2C2, C3⋊Dic3.31D4, C32⋊2SD16⋊6C2, C3⋊Dic3.9C23, D6⋊S3.2C22, C32⋊2Q8.3C22, C32⋊2C8.10C22, C2.18(C2×S3≀C2), (C3×C6).18(C2×D4), (C2×C32⋊2C8)⋊5C2, (C2×C3⋊Dic3).96C22, SmallGroup(288,885)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.13D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b4, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=b3c3 >
Subgroups: 496 in 102 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C4○D8, C3×Dic3, C3⋊Dic3, S3×C6, C62, D4⋊2S3, C32⋊2C8, S3×Dic3, D6⋊S3, C32⋊2Q8, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊D8, C32⋊2SD16, C32⋊Q16, C2×C32⋊2C8, D6.4D6, C62.13D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D8, S3≀C2, C2×S3≀C2, C62.13D4
Character table of C62.13D4
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 2 | 12 | 12 | 4 | 4 | 9 | 9 | 12 | 12 | 18 | 4 | 4 | 8 | 8 | 24 | 24 | 18 | 18 | 18 | 18 | 24 | 24 | |
| 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 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√-2 | -√2 | √-2 | √2 | 0 | 0 | complex lifted from C4○D8 |
| ρ12 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√-2 | √2 | √-2 | -√2 | 0 | 0 | complex lifted from C4○D8 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √-2 | √2 | -√-2 | -√2 | 0 | 0 | complex lifted from C4○D8 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √-2 | -√2 | -√-2 | √2 | 0 | 0 | complex lifted from C4○D8 |
| ρ15 | 4 | 4 | 4 | 0 | -2 | 1 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | 4 | -4 | 0 | -2 | 1 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 1 | 2 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S3≀C2 |
| ρ17 | 4 | 4 | -4 | -2 | 0 | -2 | 1 | 0 | 0 | 0 | 2 | 0 | 1 | -2 | -1 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ18 | 4 | 4 | 4 | 2 | 0 | -2 | 1 | 0 | 0 | 0 | 2 | 0 | 1 | -2 | 1 | -2 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | 0 | orthogonal lifted from S3≀C2 |
| ρ19 | 4 | 4 | -4 | 0 | 2 | 1 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | 1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C2×S3≀C2 |
| ρ20 | 4 | 4 | 4 | 0 | 2 | 1 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 1 | -2 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ21 | 4 | 4 | -4 | 2 | 0 | -2 | 1 | 0 | 0 | 0 | -2 | 0 | 1 | -2 | -1 | 2 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ22 | 4 | 4 | 4 | -2 | 0 | -2 | 1 | 0 | 0 | 0 | -2 | 0 | 1 | -2 | 1 | -2 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | orthogonal lifted from S3≀C2 |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 8 | -8 | 0 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 42 22 26 13 37)(2 38 14 27 23 43)(3 39 15 28 24 44)(4 45 17 29 16 40)(5 46 18 30 9 33)(6 34 10 31 19 47)(7 35 11 32 20 48)(8 41 21 25 12 36)
(1 18 13 5 22 9)(2 19 14 6 23 10)(3 11 24 7 15 20)(4 12 17 8 16 21)(25 45 36 29 41 40)(26 33 42 30 37 46)(27 34 43 31 38 47)(28 48 39 32 44 35)
(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 8)(2 7)(3 6)(4 5)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 30)(26 29)(27 28)(31 32)(33 41)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)
G:=sub<Sym(48)| (1,42,22,26,13,37)(2,38,14,27,23,43)(3,39,15,28,24,44)(4,45,17,29,16,40)(5,46,18,30,9,33)(6,34,10,31,19,47)(7,35,11,32,20,48)(8,41,21,25,12,36), (1,18,13,5,22,9)(2,19,14,6,23,10)(3,11,24,7,15,20)(4,12,17,8,16,21)(25,45,36,29,41,40)(26,33,42,30,37,46)(27,34,43,31,38,47)(28,48,39,32,44,35), (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), (1,8)(2,7)(3,6)(4,5)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,30)(26,29)(27,28)(31,32)(33,41)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)>;
G:=Group( (1,42,22,26,13,37)(2,38,14,27,23,43)(3,39,15,28,24,44)(4,45,17,29,16,40)(5,46,18,30,9,33)(6,34,10,31,19,47)(7,35,11,32,20,48)(8,41,21,25,12,36), (1,18,13,5,22,9)(2,19,14,6,23,10)(3,11,24,7,15,20)(4,12,17,8,16,21)(25,45,36,29,41,40)(26,33,42,30,37,46)(27,34,43,31,38,47)(28,48,39,32,44,35), (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), (1,8)(2,7)(3,6)(4,5)(9,17)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(25,30)(26,29)(27,28)(31,32)(33,41)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42) );
G=PermutationGroup([[(1,42,22,26,13,37),(2,38,14,27,23,43),(3,39,15,28,24,44),(4,45,17,29,16,40),(5,46,18,30,9,33),(6,34,10,31,19,47),(7,35,11,32,20,48),(8,41,21,25,12,36)], [(1,18,13,5,22,9),(2,19,14,6,23,10),(3,11,24,7,15,20),(4,12,17,8,16,21),(25,45,36,29,41,40),(26,33,42,30,37,46),(27,34,43,31,38,47),(28,48,39,32,44,35)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,17),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(25,30),(26,29),(27,28),(31,32),(33,41),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42)]])
Matrix representation of C62.13D4 ►in GL6(𝔽73)
| 27 | 54 | 0 | 0 | 0 | 0 |
| 46 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 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 |
| 41 | 41 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 41 | 41 | 0 | 0 | 0 | 0 |
| 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(73))| [27,46,0,0,0,0,54,46,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[41,16,0,0,0,0,41,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,1,0,0,0],[41,16,0,0,0,0,41,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C62.13D4 in GAP, Magma, Sage, TeX
C_6^2._{13}D_4 % in TeX
G:=Group("C6^2.13D4"); // GroupNames label
G:=SmallGroup(288,885);
// by ID
G=gap.SmallGroup(288,885);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,675,346,80,2693,2028,362,797,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^4=b^3,a*b=b*a,c*a*c^-1=a^3*b^4,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=b^3*c^3>;
// generators/relations
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