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, C22.27(C2×S4), C2.10(C22×S4), Q8.2(C22×S3), (C2×GL2(𝔽3))⋊1C2, (C2×CSU2(𝔽3))⋊4C2, (C22×SL2(𝔽3))⋊6C2, (C2×SL2(𝔽3))⋊5C22, SmallGroup(192,1476)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C2×Q8.D6 |
Generators and relations for C2×Q8.D6
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 >
Subgroups: 555 in 153 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C8, C2×C4, D4, Q8, Q8, C23, C23, Dic3, D6, C2×C6, C2×C8, M4(2), SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, SL2(𝔽3), C2×Dic3, C3⋊D4, C22×S3, C22×C6, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C2×SL2(𝔽3), C2×C3⋊D4, C2×C8.C22, C2×CSU2(𝔽3), C2×GL2(𝔽3), Q8.D6, C22×SL2(𝔽3), C2×Q8.D6
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, Q8.D6, C22×S4, C2×Q8.D6
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 |
►On 32 points
Generators in S32
(1 6)(2 5)(3 4)(7 8)(9 19)(10 20)(11 15)(12 16)(13 17)(14 18)(21 24)(22 25)(23 26)(27 30)(28 31)(29 32)
(1 12 5 19)(2 9 6 16)(3 26 8 29)(4 23 7 32)(10 15 17 14)(11 13 18 20)(21 28 30 25)(22 24 31 27)
(1 10 5 17)(2 13 6 20)(3 24 8 27)(4 21 7 30)(9 11 16 18)(12 14 19 15)(22 29 31 26)(23 25 32 28)
(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 8 5 3)(2 4 6 7)(9 32 16 23)(10 22 17 31)(11 30 18 21)(12 26 19 29)(13 28 20 25)(14 24 15 27)
G:=sub<Sym(32)| (1,6)(2,5)(3,4)(7,8)(9,19)(10,20)(11,15)(12,16)(13,17)(14,18)(21,24)(22,25)(23,26)(27,30)(28,31)(29,32), (1,12,5,19)(2,9,6,16)(3,26,8,29)(4,23,7,32)(10,15,17,14)(11,13,18,20)(21,28,30,25)(22,24,31,27), (1,10,5,17)(2,13,6,20)(3,24,8,27)(4,21,7,30)(9,11,16,18)(12,14,19,15)(22,29,31,26)(23,25,32,28), (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), (1,8,5,3)(2,4,6,7)(9,32,16,23)(10,22,17,31)(11,30,18,21)(12,26,19,29)(13,28,20,25)(14,24,15,27)>;
G:=Group( (1,6)(2,5)(3,4)(7,8)(9,19)(10,20)(11,15)(12,16)(13,17)(14,18)(21,24)(22,25)(23,26)(27,30)(28,31)(29,32), (1,12,5,19)(2,9,6,16)(3,26,8,29)(4,23,7,32)(10,15,17,14)(11,13,18,20)(21,28,30,25)(22,24,31,27), (1,10,5,17)(2,13,6,20)(3,24,8,27)(4,21,7,30)(9,11,16,18)(12,14,19,15)(22,29,31,26)(23,25,32,28), (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), (1,8,5,3)(2,4,6,7)(9,32,16,23)(10,22,17,31)(11,30,18,21)(12,26,19,29)(13,28,20,25)(14,24,15,27) );
G=PermutationGroup([[(1,6),(2,5),(3,4),(7,8),(9,19),(10,20),(11,15),(12,16),(13,17),(14,18),(21,24),(22,25),(23,26),(27,30),(28,31),(29,32)], [(1,12,5,19),(2,9,6,16),(3,26,8,29),(4,23,7,32),(10,15,17,14),(11,13,18,20),(21,28,30,25),(22,24,31,27)], [(1,10,5,17),(2,13,6,20),(3,24,8,27),(4,21,7,30),(9,11,16,18),(12,14,19,15),(22,29,31,26),(23,25,32,28)], [(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)], [(1,8,5,3),(2,4,6,7),(9,32,16,23),(10,22,17,31),(11,30,18,21),(12,26,19,29),(13,28,20,25),(14,24,15,27)]])
Matrix representation of C2×Q8.D6 ►in 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] >;
C2×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
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