non-abelian, soluble, monomial
Aliases: 2+ 1+4.3C6, C4○D4⋊2A4, (C22×C4)⋊3A4, Q8.2(C2×A4), C23⋊A4⋊4C2, C23.7(C2×A4), C4.2(C22⋊A4), C2.C25⋊2C3, C2.5(C2×C22⋊A4), SmallGroup(192,1509)
Series: Derived ►Chief ►Lower central ►Upper central
| 2+ 1+4 — 2+ 1+4.3C6 |
Generators and relations for 2+ 1+4.3C6
G = < a,b,c,d,e | a4=b2=d2=1, c2=e6=a2, bab=a-1, ac=ca, ad=da, eae-1=a2bc, bc=cb, bd=db, ebe-1=a2bcd, dcd=a2c, ece-1=a-1d, ede-1=a-1bd >
Subgroups: 567 in 165 conjugacy classes, 17 normal (7 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, D4, Q8, Q8, C23, C23, C12, A4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C2×A4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, C4×A4, C4.A4, C2.C25, C23⋊A4, 2+ 1+4.3C6
Quotients: C1, C2, C3, C6, A4, C2×A4, C22⋊A4, C2×C22⋊A4, 2+ 1+4.3C6
Character table of 2+ 1+4.3C6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 16 | 16 | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ9 | 3 | 3 | 1 | -1 | -1 | -1 | -3 | 0 | 0 | -3 | -3 | 3 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ10 | 3 | 3 | -3 | -1 | -1 | -1 | 1 | 0 | 0 | -3 | -3 | -1 | 1 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ11 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ12 | 3 | 3 | 1 | -1 | 3 | -1 | 1 | 0 | 0 | -3 | -3 | -1 | 1 | 1 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ13 | 3 | 3 | 1 | -1 | -1 | 3 | 1 | 0 | 0 | -3 | -3 | -1 | -3 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 1 | 3 | -1 | -1 | 1 | 0 | 0 | -3 | -3 | -1 | 1 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ16 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | -1 | -1 | i | i | -i | -i | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -i | -i | i | i | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | complex faithful |
►On 16 points - transitive group 16T423
(1 16 3 10)(2 13 4 7)(5 15 11 9)(6 14 12 8)
(1 10)(2 7)(3 16)(4 13)(5 9)(6 14)(8 12)(11 15)
(1 14 3 8)(2 11 4 5)(6 16 12 10)(7 15 13 9)
(5 11)(6 12)(8 14)(9 15)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,16,3,10)(2,13,4,7)(5,15,11,9)(6,14,12,8), (1,10)(2,7)(3,16)(4,13)(5,9)(6,14)(8,12)(11,15), (1,14,3,8)(2,11,4,5)(6,16,12,10)(7,15,13,9), (5,11)(6,12)(8,14)(9,15), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (1,16,3,10)(2,13,4,7)(5,15,11,9)(6,14,12,8), (1,10)(2,7)(3,16)(4,13)(5,9)(6,14)(8,12)(11,15), (1,14,3,8)(2,11,4,5)(6,16,12,10)(7,15,13,9), (5,11)(6,12)(8,14)(9,15), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,16,3,10),(2,13,4,7),(5,15,11,9),(6,14,12,8)], [(1,10),(2,7),(3,16),(4,13),(5,9),(6,14),(8,12),(11,15)], [(1,14,3,8),(2,11,4,5),(6,16,12,10),(7,15,13,9)], [(5,11),(6,12),(8,14),(9,15)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,423);
Matrix representation of 2+ 1+4.3C6 ►in GL4(𝔽5) generated by
| 0 | 3 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 |
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 |
| 2 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,3,0,0,3,0,0,0,0,0,0,2,0,0,2,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[0,0,3,0,0,0,0,2,3,0,0,0,0,2,0,0],[0,0,2,0,0,0,0,3,3,0,0,0,0,2,0,0],[2,0,0,0,0,0,0,1,0,3,0,0,0,0,1,0] >;
2+ 1+4.3C6 in GAP, Magma, Sage, TeX
2_+^{1+4}._3C_6 % in TeX
G:=Group("ES+(2,2).3C6"); // GroupNames label
G:=SmallGroup(192,1509);
// by ID
G=gap.SmallGroup(192,1509);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,262,851,375,1524,1027]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=d^2=1,c^2=e^6=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^2*b*c,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*b*c*d,d*c*d=a^2*c,e*c*e^-1=a^-1*d,e*d*e^-1=a^-1*b*d>;
// generators/relations
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