direct product, non-abelian, soluble
Aliases: C22×SL2(𝔽3), C23.6A4, Q8⋊(C2×C6), (C2×Q8)⋊2C6, (C22×Q8)⋊1C3, C2.3(C22×A4), C22.8(C2×A4), SmallGroup(96,198)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C22×SL2(𝔽3) |
| Q8 — C22×SL2(𝔽3) |
Generators and relations for C22×SL2(𝔽3)
G = < a,b,c,d,e | a2=b2=c4=e3=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Subgroups: 147 in 61 conjugacy classes, 26 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, Q8, Q8, C23, C2×C6, C22×C4, C2×Q8, C2×Q8, SL2(𝔽3), C22×C6, C22×Q8, C2×SL2(𝔽3), C22×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, SL2(𝔽3), C2×A4, C2×SL2(𝔽3), C22×A4, C22×SL2(𝔽3)
Character table of C22×SL2(𝔽3)
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | -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 | -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 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | 1 | -1 | ζ32 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ6 | ζ6 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | -1 | 1 | 1 | -1 | ζ65 | ζ6 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | linear of order 6 |
| ρ7 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | 1 | -1 | ζ3 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ65 | ζ65 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | ζ6 | ζ3 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ32 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ11 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | ζ65 | ζ32 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ3 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | -1 | 1 | 1 | -1 | ζ6 | ζ65 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | linear of order 6 |
| ρ13 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ14 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ15 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ3 | ζ6 | ζ32 | ζ6 | ζ32 | ζ65 | ζ3 | ζ3 | ζ65 | ζ3 | ζ6 | ζ32 | ζ65 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ18 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ3 | ζ3 | ζ65 | ζ32 | ζ6 | ζ32 | ζ6 | ζ32 | ζ3 | ζ65 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ19 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ65 | ζ32 | ζ6 | ζ6 | ζ32 | ζ65 | ζ3 | ζ3 | ζ3 | ζ65 | ζ32 | ζ6 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ20 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ32 | ζ65 | ζ3 | ζ65 | ζ3 | ζ6 | ζ32 | ζ32 | ζ6 | ζ32 | ζ65 | ζ3 | ζ6 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ32 | ζ32 | ζ6 | ζ3 | ζ65 | ζ3 | ζ65 | ζ3 | ζ32 | ζ6 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ22 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ65 | ζ3 | ζ65 | ζ32 | ζ6 | ζ32 | ζ32 | ζ6 | ζ65 | ζ3 | ζ6 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ23 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ6 | ζ32 | ζ6 | ζ3 | ζ65 | ζ3 | ζ3 | ζ65 | ζ6 | ζ32 | ζ65 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ24 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ6 | ζ3 | ζ65 | ζ65 | ζ3 | ζ6 | ζ32 | ζ32 | ζ32 | ζ6 | ζ3 | ζ65 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ25 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ26 | 3 | -3 | -3 | -3 | 3 | -3 | 3 | 3 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ27 | 3 | 3 | -3 | -3 | 3 | 3 | -3 | -3 | 0 | 0 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ28 | 3 | -3 | 3 | 3 | 3 | -3 | -3 | -3 | 0 | 0 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
►On 32 points
Generators in S32
(1 19)(2 20)(3 17)(4 18)(5 23)(6 24)(7 21)(8 22)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)
(1 11)(2 12)(3 9)(4 10)(5 15)(6 16)(7 13)(8 14)(17 27)(18 28)(19 25)(20 26)(21 31)(22 32)(23 29)(24 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 7 3 5)(2 6 4 8)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)
(2 6 7)(4 8 5)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)
G:=sub<Sym(32)| (1,19)(2,20)(3,17)(4,18)(5,23)(6,24)(7,21)(8,22)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30), (1,11)(2,12)(3,9)(4,10)(5,15)(6,16)(7,13)(8,14)(17,27)(18,28)(19,25)(20,26)(21,31)(22,32)(23,29)(24,30), (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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32), (2,6,7)(4,8,5)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29)>;
G:=Group( (1,19)(2,20)(3,17)(4,18)(5,23)(6,24)(7,21)(8,22)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30), (1,11)(2,12)(3,9)(4,10)(5,15)(6,16)(7,13)(8,14)(17,27)(18,28)(19,25)(20,26)(21,31)(22,32)(23,29)(24,30), (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), (1,7,3,5)(2,6,4,8)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32), (2,6,7)(4,8,5)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29) );
G=PermutationGroup([[(1,19),(2,20),(3,17),(4,18),(5,23),(6,24),(7,21),(8,22),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30)], [(1,11),(2,12),(3,9),(4,10),(5,15),(6,16),(7,13),(8,14),(17,27),(18,28),(19,25),(20,26),(21,31),(22,32),(23,29),(24,30)], [(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)], [(1,7,3,5),(2,6,4,8),(9,15,11,13),(10,14,12,16),(17,23,19,21),(18,22,20,24),(25,31,27,29),(26,30,28,32)], [(2,6,7),(4,8,5),(10,14,15),(12,16,13),(18,22,23),(20,24,21),(26,30,31),(28,32,29)]])
C22×SL2(𝔽3) is a maximal subgroup of
C23.14S4 C23.15S4 C23.16S4 (C2×Q8)⋊C12 SL2(𝔽3)⋊5D4
Matrix representation of C22×SL2(𝔽3) ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 4 |
| 0 | 0 | 4 | 10 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 10 | 9 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,3,4,0,0,4,10],[9,0,0,0,0,9,0,0,0,0,1,10,0,0,0,9] >;
C22×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_2^2\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C2^2xSL(2,3)"); // GroupNames label
G:=SmallGroup(96,198);
// by ID
G=gap.SmallGroup(96,198);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,159,117,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^3=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations
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