non-abelian, soluble, monomial
Aliases: C62.15D4, C32⋊Q16⋊4C2, C22.6S3≀C2, C3⋊Dic3.33D4, C62.C4⋊2C2, D6.4D6.2C2, C32⋊2SD16⋊2C2, C32⋊2(C8.C22), D6⋊S3.3C22, C3⋊Dic3.11C23, C32⋊2C8.2C22, C32⋊2Q8.7C22, C2.20(C2×S3≀C2), (C3×C6).20(C2×D4), (C2×C32⋊2Q8)⋊10C2, (C2×C3⋊Dic3).98C22, SmallGroup(288,887)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.15D4
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=c3 >
Subgroups: 464 in 99 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, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C8.C22, C3×Dic3, C3⋊Dic3, S3×C6, C62, C2×Dic6, D4⋊2S3, C32⋊2C8, S3×Dic3, D6⋊S3, C32⋊2Q8, C32⋊2Q8, C32⋊2Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊2SD16, C32⋊Q16, C62.C4, D6.4D6, C2×C32⋊2Q8, C62.15D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C8.C22, S3≀C2, C2×S3≀C2, C62.15D4
Character table of C62.15D4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | |
| size | 1 | 1 | 2 | 12 | 4 | 4 | 12 | 12 | 12 | 18 | 18 | 4 | 4 | 4 | 4 | 8 | 24 | 36 | 36 | 12 | 12 | 12 | 12 | 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 | 2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 4 | 2 | -2 | 1 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ12 | 4 | 4 | -4 | -2 | -2 | 1 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | 2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S3≀C2 |
| ρ13 | 4 | 4 | -4 | 0 | 1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | -1 | -2 | 2 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ14 | 4 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ15 | 4 | 4 | 4 | 0 | 1 | -2 | 2 | 2 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | 4 | 4 | 0 | 1 | -2 | -2 | -2 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 4 | -4 | 2 | -2 | 1 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | 2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from C2×S3≀C2 |
| ρ18 | 4 | 4 | -4 | 0 | 1 | -2 | 2 | -2 | 0 | 0 | 0 | -1 | 1 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 0 | orthogonal lifted from C2×S3≀C2 |
| ρ19 | 4 | -4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ20 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | -3 | 2 | 0 | 0 | 0 | 0 | -√3 | √3 | -√3 | √3 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | -3 | 2 | 0 | 0 | 0 | 0 | √3 | -√3 | √3 | -√3 | 0 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | -3 | -1 | 3 | 2 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | -√3 | 0 | symplectic faithful, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | -3 | -1 | 3 | 2 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | √3 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 8 | -8 | 0 | 0 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 29 35 13 24 46)(2 10)(3 48 18 15 37 31)(4 12)(5 25 39 9 20 42)(6 14)(7 44 22 11 33 27)(8 16)(17 26)(19 28)(21 30)(23 32)(34 41)(36 43)(38 45)(40 47)
(1 5)(2 40 17 6 36 21)(3 7)(4 23 38 8 19 34)(9 13)(10 47 26 14 43 30)(11 15)(12 32 45 16 28 41)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 38)(18 33)(19 36)(20 39)(21 34)(22 37)(23 40)(24 35)(25 46)(26 41)(27 44)(28 47)(29 42)(30 45)(31 48)(32 43)
G:=sub<Sym(48)| (1,29,35,13,24,46)(2,10)(3,48,18,15,37,31)(4,12)(5,25,39,9,20,42)(6,14)(7,44,22,11,33,27)(8,16)(17,26)(19,28)(21,30)(23,32)(34,41)(36,43)(38,45)(40,47), (1,5)(2,40,17,6,36,21)(3,7)(4,23,38,8,19,34)(9,13)(10,47,26,14,43,30)(11,15)(12,32,45,16,28,41)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43)>;
G:=Group( (1,29,35,13,24,46)(2,10)(3,48,18,15,37,31)(4,12)(5,25,39,9,20,42)(6,14)(7,44,22,11,33,27)(8,16)(17,26)(19,28)(21,30)(23,32)(34,41)(36,43)(38,45)(40,47), (1,5)(2,40,17,6,36,21)(3,7)(4,23,38,8,19,34)(9,13)(10,47,26,14,43,30)(11,15)(12,32,45,16,28,41)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,38)(18,33)(19,36)(20,39)(21,34)(22,37)(23,40)(24,35)(25,46)(26,41)(27,44)(28,47)(29,42)(30,45)(31,48)(32,43) );
G=PermutationGroup([[(1,29,35,13,24,46),(2,10),(3,48,18,15,37,31),(4,12),(5,25,39,9,20,42),(6,14),(7,44,22,11,33,27),(8,16),(17,26),(19,28),(21,30),(23,32),(34,41),(36,43),(38,45),(40,47)], [(1,5),(2,40,17,6,36,21),(3,7),(4,23,38,8,19,34),(9,13),(10,47,26,14,43,30),(11,15),(12,32,45,16,28,41),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,38),(18,33),(19,36),(20,39),(21,34),(22,37),(23,40),(24,35),(25,46),(26,41),(27,44),(28,47),(29,42),(30,45),(31,48),(32,43)]])
Matrix representation of C62.15D4 ►in GL8(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 46 | 19 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(73))| [72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,19,27,0,0,0,0,0,0,0,0,46,27,0,0,0,0,0,0,19,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,71,1,0,0],[1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,71,1] >;
C62.15D4 in GAP, Magma, Sage, TeX
C_6^2._{15}D_4 % in TeX
G:=Group("C6^2.15D4"); // GroupNames label
G:=SmallGroup(288,887);
// by ID
G=gap.SmallGroup(288,887);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,675,346,80,2693,2028,362,797,1203]);
// 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,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=c^3>;
// generators/relations
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