direct product, non-abelian, soluble
Aliases: C2×Q8.D6, C23.18S4, GL2(𝔽3)⋊1C22, CSU2(𝔽3)⋊1C22, SL2(𝔽3).2C23, (C2×Q8)⋊3D6, (C22×Q8)⋊4S3, C2.10(C22×S4), C22.27(C2×S4), Q8.2(C22×S3), (C2×GL2(𝔽3))⋊1C2, (C2×CSU2(𝔽3))⋊4C2, (C2×SL2(𝔽3))⋊5C22, (C22×SL2(𝔽3))⋊6C2, SmallGroup(192,1476)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C2×Q8.D6 |
Subgroups: 555 in 153 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×6], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×5], C8 [×4], C2×C4 [×11], D4 [×7], Q8, Q8 [×7], C23, C23, Dic3 [×2], D6 [×4], C2×C6 [×5], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×4], C4○D4 [×6], SL2(𝔽3), C2×Dic3, C3⋊D4 [×4], C22×S3, C22×C6, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, CSU2(𝔽3) [×2], GL2(𝔽3) [×2], C2×SL2(𝔽3), C2×SL2(𝔽3) [×2], C2×C3⋊D4, C2×C8.C22, C2×CSU2(𝔽3), C2×GL2(𝔽3), Q8.D6 [×4], C22×SL2(𝔽3), C2×Q8.D6
Quotients:
C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], Q8.D6 [×2], C22×S4, C2×Q8.D6
Generators and relations
G = < a,b,c,d,e | a2=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, dbd-1=bc, dcd-1=b, ece-1=b-1c, ede-1=b2d-1 >
►On 32 points
(1 8)(2 7)(3 4)(5 6)(9 12)(10 13)(11 14)(15 28)(16 29)(17 30)(18 31)(19 32)(20 27)(21 24)(22 25)(23 26)
(1 28 7 18)(2 31 8 15)(3 23 6 13)(4 26 5 10)(9 11 25 21)(12 14 22 24)(16 30 32 20)(17 19 27 29)
(1 32 7 16)(2 29 8 19)(3 21 6 11)(4 24 5 14)(9 23 25 13)(10 12 26 22)(15 17 31 27)(18 20 28 30)
(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)
(1 6 7 3)(2 4 8 5)(9 32 25 16)(10 15 26 31)(11 30 21 20)(12 19 22 29)(13 28 23 18)(14 17 24 27)
G:=sub<Sym(32)| (1,8)(2,7)(3,4)(5,6)(9,12)(10,13)(11,14)(15,28)(16,29)(17,30)(18,31)(19,32)(20,27)(21,24)(22,25)(23,26), (1,28,7,18)(2,31,8,15)(3,23,6,13)(4,26,5,10)(9,11,25,21)(12,14,22,24)(16,30,32,20)(17,19,27,29), (1,32,7,16)(2,29,8,19)(3,21,6,11)(4,24,5,14)(9,23,25,13)(10,12,26,22)(15,17,31,27)(18,20,28,30), (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), (1,6,7,3)(2,4,8,5)(9,32,25,16)(10,15,26,31)(11,30,21,20)(12,19,22,29)(13,28,23,18)(14,17,24,27)>;
G:=Group( (1,8)(2,7)(3,4)(5,6)(9,12)(10,13)(11,14)(15,28)(16,29)(17,30)(18,31)(19,32)(20,27)(21,24)(22,25)(23,26), (1,28,7,18)(2,31,8,15)(3,23,6,13)(4,26,5,10)(9,11,25,21)(12,14,22,24)(16,30,32,20)(17,19,27,29), (1,32,7,16)(2,29,8,19)(3,21,6,11)(4,24,5,14)(9,23,25,13)(10,12,26,22)(15,17,31,27)(18,20,28,30), (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), (1,6,7,3)(2,4,8,5)(9,32,25,16)(10,15,26,31)(11,30,21,20)(12,19,22,29)(13,28,23,18)(14,17,24,27) );
G=PermutationGroup([(1,8),(2,7),(3,4),(5,6),(9,12),(10,13),(11,14),(15,28),(16,29),(17,30),(18,31),(19,32),(20,27),(21,24),(22,25),(23,26)], [(1,28,7,18),(2,31,8,15),(3,23,6,13),(4,26,5,10),(9,11,25,21),(12,14,22,24),(16,30,32,20),(17,19,27,29)], [(1,32,7,16),(2,29,8,19),(3,21,6,11),(4,24,5,14),(9,23,25,13),(10,12,26,22),(15,17,31,27),(18,20,28,30)], [(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)], [(1,6,7,3),(2,4,8,5),(9,32,25,16),(10,15,26,31),(11,30,21,20),(12,19,22,29),(13,28,23,18),(14,17,24,27)])
Matrix representation ►G ⊆ GL7(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 65 | 0 | 0 |
| 0 | 0 | 0 | 65 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 68 | 5 | 64 | 8 |
| 0 | 0 | 0 | 12 | 5 | 8 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 65 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 69 | 4 | 65 | 64 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 71 | 0 |
| 0 | 0 | 0 | 16 | 1 | 16 | 18 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 64 | 8 | 0 | 72 |
G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,72,1,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,9,65,0,0,0,0,0,65,64,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0],[0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,68,12,0,0,0,72,0,5,5,0,0,0,0,0,64,8,0,0,0,0,0,8,9],[1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,9,1,69,0,0,0,0,65,0,4,0,0,0,0,0,1,65,0,0,0,0,0,0,64],[72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,16,1,64,0,0,0,0,1,0,8,0,0,0,71,16,1,0,0,0,0,0,18,0,72] >;
Character table of C2×Q8.D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 8 | 6 | 6 | 6 | 6 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ13 | 3 | 3 | 3 | 3 | -3 | -3 | 1 | 1 | 0 | -1 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | -3 | -3 | 3 | 3 | -3 | 1 | -1 | 0 | 1 | -1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | -3 | -3 | 3 | -3 | 3 | 1 | -1 | 0 | 1 | 1 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ17 | 3 | -3 | -3 | 3 | -3 | 3 | -1 | 1 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | 3 | 3 | -3 | -3 | -1 | -1 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | -3 | -3 | 3 | 3 | -3 | -1 | 1 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | √-3 | √-3 | -1 | -1 | √-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | √-3 | √-3 | -1 | -1 | √-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | √-3 | √-3 | 1 | -1 | √-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | √-3 | √-3 | 1 | -1 | √-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from Q8.D6 |
In GAP, Magma, Sage, TeX
C_2\times Q_8.D_6
% in TeX
G:=Group("C2xQ8.D6"); // GroupNames label
G:=SmallGroup(192,1476);
// by ID
G=gap.SmallGroup(192,1476);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,2102,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b,e*c*e^-1=b^-1*c,e*d*e^-1=b^2*d^-1>;
// generators/relations