direct product, non-abelian, soluble
Aliases: C2×C4.A4, SL2(𝔽3)⋊3C22, C4○(C4.A4), C4○D4⋊2C6, C4.6(C2×A4), (C2×C4).2A4, Q8.1(C2×C6), (C2×Q8).2C6, C22.9(C2×A4), C2.5(C22×A4), (C2×C4)○SL2(𝔽3), C4○(C2×SL2(𝔽3)), (C2×SL2(𝔽3))⋊4C2, (C2×C4○D4)⋊C3, (C2×C4)○(C2×SL2(𝔽3)), SmallGroup(96,200)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×C4.A4 |
| Q8 — C2×C4.A4 |
Generators and relations for C2×C4.A4
G = < a,b,c,d,e | a2=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >
Character table of C2×C4.A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 4 | 4 | 1 | 1 | 1 | 1 | 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 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | -1 | 1 | -1 | 1 | -1 | ζ32 | ζ65 | ζ65 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ8 | 1 | -1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -1 | 1 | -1 | 1 | 1 | -1 | ζ32 | ζ65 | ζ65 | ζ3 | ζ6 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -1 | 1 | -1 | 1 | 1 | -1 | ζ3 | ζ6 | ζ6 | ζ32 | ζ65 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | -1 | 1 | -1 | 1 | -1 | ζ3 | ζ6 | ζ6 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | 2i | -2i | -2i | 2i | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -i | i | -i | i | i | -i | i | -i | complex lifted from C4.A4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -2i | -2i | 2i | 2i | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | complex lifted from C4.A4 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -2i | 2i | 2i | -2i | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | i | -i | i | -i | -i | i | -i | i | complex lifted from C4.A4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | 2i | 2i | -2i | -2i | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | complex lifted from C4.A4 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | ζ6 | ζ65 | 2i | -2i | -2i | 2i | 0 | 0 | ζ3 | ζ32 | ζ6 | ζ32 | ζ3 | ζ65 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | complex lifted from C4.A4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | ζ65 | ζ6 | -2i | -2i | 2i | 2i | 0 | 0 | ζ32 | ζ65 | ζ3 | ζ3 | ζ6 | ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | complex lifted from C4.A4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | ζ65 | ζ6 | 2i | -2i | -2i | 2i | 0 | 0 | ζ32 | ζ3 | ζ65 | ζ3 | ζ32 | ζ6 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | complex lifted from C4.A4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | ζ6 | ζ65 | -2i | 2i | 2i | -2i | 0 | 0 | ζ3 | ζ32 | ζ6 | ζ32 | ζ3 | ζ65 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | complex lifted from C4.A4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | ζ6 | ζ65 | 2i | 2i | -2i | -2i | 0 | 0 | ζ3 | ζ6 | ζ32 | ζ32 | ζ65 | ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | complex lifted from C4.A4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | ζ65 | ζ6 | -2i | 2i | 2i | -2i | 0 | 0 | ζ32 | ζ3 | ζ65 | ζ3 | ζ32 | ζ6 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | complex lifted from C4.A4 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | ζ65 | ζ6 | 2i | 2i | -2i | -2i | 0 | 0 | ζ32 | ζ65 | ζ3 | ζ3 | ζ6 | ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | complex lifted from C4.A4 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | ζ6 | ζ65 | -2i | -2i | 2i | 2i | 0 | 0 | ζ3 | ζ6 | ζ32 | ζ32 | ζ65 | ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | complex lifted from C4.A4 |
| ρ25 | 3 | -3 | 3 | -3 | -1 | 1 | 0 | 0 | 3 | -3 | 3 | -3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ26 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ27 | 3 | -3 | 3 | -3 | 1 | -1 | 0 | 0 | -3 | 3 | -3 | 3 | -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 | 1 | 1 | 0 | 0 | -3 | -3 | -3 | -3 | -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 13)(2 14)(3 15)(4 16)(5 26)(6 27)(7 28)(8 25)(9 17)(10 18)(11 19)(12 20)(21 32)(22 29)(23 30)(24 31)
(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 27 3 25)(2 28 4 26)(5 14 7 16)(6 15 8 13)(9 31 11 29)(10 32 12 30)(17 24 19 22)(18 21 20 23)
(1 29 3 31)(2 30 4 32)(5 18 7 20)(6 19 8 17)(9 27 11 25)(10 28 12 26)(13 22 15 24)(14 23 16 21)
(5 21 20)(6 22 17)(7 23 18)(8 24 19)(9 27 29)(10 28 30)(11 25 31)(12 26 32)
G:=sub<Sym(32)| (1,13)(2,14)(3,15)(4,16)(5,26)(6,27)(7,28)(8,25)(9,17)(10,18)(11,19)(12,20)(21,32)(22,29)(23,30)(24,31), (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), (1,27,3,25)(2,28,4,26)(5,14,7,16)(6,15,8,13)(9,31,11,29)(10,32,12,30)(17,24,19,22)(18,21,20,23), (1,29,3,31)(2,30,4,32)(5,18,7,20)(6,19,8,17)(9,27,11,25)(10,28,12,26)(13,22,15,24)(14,23,16,21), (5,21,20)(6,22,17)(7,23,18)(8,24,19)(9,27,29)(10,28,30)(11,25,31)(12,26,32)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,26)(6,27)(7,28)(8,25)(9,17)(10,18)(11,19)(12,20)(21,32)(22,29)(23,30)(24,31), (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), (1,27,3,25)(2,28,4,26)(5,14,7,16)(6,15,8,13)(9,31,11,29)(10,32,12,30)(17,24,19,22)(18,21,20,23), (1,29,3,31)(2,30,4,32)(5,18,7,20)(6,19,8,17)(9,27,11,25)(10,28,12,26)(13,22,15,24)(14,23,16,21), (5,21,20)(6,22,17)(7,23,18)(8,24,19)(9,27,29)(10,28,30)(11,25,31)(12,26,32) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,26),(6,27),(7,28),(8,25),(9,17),(10,18),(11,19),(12,20),(21,32),(22,29),(23,30),(24,31)], [(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)], [(1,27,3,25),(2,28,4,26),(5,14,7,16),(6,15,8,13),(9,31,11,29),(10,32,12,30),(17,24,19,22),(18,21,20,23)], [(1,29,3,31),(2,30,4,32),(5,18,7,20),(6,19,8,17),(9,27,11,25),(10,28,12,26),(13,22,15,24),(14,23,16,21)], [(5,21,20),(6,22,17),(7,23,18),(8,24,19),(9,27,29),(10,28,30),(11,25,31),(12,26,32)]])
C2×C4.A4 is a maximal subgroup of
U2(𝔽3)⋊C2 C4.A4⋊C4 SL2(𝔽3).D4 (C2×C4).S4 SL2(𝔽3)⋊D4 C4○D4⋊C12 SL2(𝔽3)⋊5D4 SL2(𝔽3)⋊6D4 M4(2).A4 GL2(𝔽3)⋊C22 2- 1+4⋊3C6
C2×C4.A4 is a maximal quotient of
C2×C4×SL2(𝔽3) SL2(𝔽3)⋊5D4 SL2(𝔽3)⋊6D4 SL2(𝔽3)⋊3Q8
Matrix representation of C2×C4.A4 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 9 | 3 |
| 0 | 3 | 4 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 9 | 3 |
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[1,0,0,0,5,0,0,0,5],[1,0,0,0,9,3,0,3,4],[1,0,0,0,0,12,0,1,0],[3,0,0,0,1,9,0,0,3] >;
C2×C4.A4 in GAP, Magma, Sage, TeX
C_2\times C_4.A_4
% in TeX
G:=Group("C2xC4.A4"); // GroupNames label
G:=SmallGroup(96,200);
// by ID
G=gap.SmallGroup(96,200);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,295,159,117,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^3=1,c^2=d^2=b^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=b^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of C2×C4.A4 in TeX
Character table of C2×C4.A4 in TeX