non-abelian, soluble, monomial
Aliases: C24.10D6, C4⋊S4⋊5C2, (C2×C4)⋊3S4, (C4×S4)⋊4C2, A4⋊Q8⋊5C2, C4.34(C2×S4), (C23×C4)⋊4S3, A4⋊1(C4○D4), A4⋊D4⋊3C2, C22⋊(C4○D12), C2.5(C22×S4), C22.7(C2×S4), A4⋊C4.2C22, (C2×A4).4C23, (C2×S4).1C22, (C22×C4).12D6, (C4×A4).16C22, C23.4(C22×S3), (C22×A4).11C22, (C2×C4×A4)⋊4C2, SmallGroup(192,1471)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.10D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=b, faf-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, fcf-1=ede-1=cd=dc, ece-1=d, df=fd, fef-1=e5 >
Subgroups: 626 in 171 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, A4, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, S4, C2×A4, C2×A4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, A4⋊C4, C4×A4, C4○D12, C2×S4, C22×A4, C22.19C24, A4⋊Q8, C4×S4, C4⋊S4, A4⋊D4, C2×C4×A4, C24.10D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, S4, C22×S3, C4○D12, C2×S4, C22×S4, C24.10D6
Character table of C24.10D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 12 | 12 | 8 | 1 | 1 | 2 | 3 | 3 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | -1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | -2i | 2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2i | -2i | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 2i | -2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2i | 2i | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-3 | √-3 | 1 | √3 | i | -i | -√3 | complex lifted from C4○D12 |
| ρ16 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | 2i | -2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-3 | -√-3 | 1 | -√3 | i | -i | √3 | complex lifted from C4○D12 |
| ρ17 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-3 | -√-3 | 1 | √3 | -i | i | -√3 | complex lifted from C4○D12 |
| ρ18 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | -2i | 2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-3 | √-3 | 1 | -√3 | -i | i | √3 | complex lifted from C4○D12 |
| ρ19 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 0 | -3 | -3 | -3 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | 3 | -3 | -1 | -1 | 1 | -1 | 1 | 0 | 3 | 3 | -3 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 3 | 3 | -3 | -1 | -1 | 1 | 1 | -1 | 0 | 3 | 3 | -3 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ22 | 3 | 3 | 3 | -1 | -1 | -1 | 1 | -1 | 0 | -3 | -3 | -3 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 3 | 3 | 3 | -1 | -1 | -1 | 1 | 1 | 0 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ24 | 3 | 3 | -3 | -1 | -1 | 1 | 1 | 1 | 0 | -3 | -3 | 3 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ25 | 3 | 3 | -3 | -1 | -1 | 1 | -1 | -1 | 0 | -3 | -3 | 3 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | 3 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ27 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -6i | 6i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 6i | -6i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T292
Generators in S24
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(2 8)(3 9)(5 11)(6 12)(13 19)(15 21)(16 22)(18 24)
(1 7)(2 8)(4 10)(5 11)(14 20)(15 21)(17 23)(18 24)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 18 7 24)(2 23 8 17)(3 16 9 22)(4 21 10 15)(5 14 11 20)(6 19 12 13)
G:=sub<Sym(24)| (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,8)(3,9)(5,11)(6,12)(13,19)(15,21)(16,22)(18,24), (1,7)(2,8)(4,10)(5,11)(14,20)(15,21)(17,23)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18,7,24)(2,23,8,17)(3,16,9,22)(4,21,10,15)(5,14,11,20)(6,19,12,13)>;
G:=Group( (13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (2,8)(3,9)(5,11)(6,12)(13,19)(15,21)(16,22)(18,24), (1,7)(2,8)(4,10)(5,11)(14,20)(15,21)(17,23)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18,7,24)(2,23,8,17)(3,16,9,22)(4,21,10,15)(5,14,11,20)(6,19,12,13) );
G=PermutationGroup([[(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(2,8),(3,9),(5,11),(6,12),(13,19),(15,21),(16,22),(18,24)], [(1,7),(2,8),(4,10),(5,11),(14,20),(15,21),(17,23),(18,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18,7,24),(2,23,8,17),(3,16,9,22),(4,21,10,15),(5,14,11,20),(6,19,12,13)]])
G:=TransitiveGroup(24,292);
►On 24 points - transitive group 24T319
Generators in S24
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 7)(2 8)(4 10)(5 11)(14 20)(15 21)(17 23)(18 24)
(1 7)(3 9)(4 10)(6 12)(13 19)(14 20)(16 22)(17 23)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)
G:=sub<Sym(24)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(4,10)(5,11)(14,20)(15,21)(17,23)(18,24), (1,7)(3,9)(4,10)(6,12)(13,19)(14,20)(16,22)(17,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,7)(2,8)(4,10)(5,11)(14,20)(15,21)(17,23)(18,24), (1,7)(3,9)(4,10)(6,12)(13,19)(14,20)(16,22)(17,23), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17) );
G=PermutationGroup([[(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,7),(2,8),(4,10),(5,11),(14,20),(15,21),(17,23),(18,24)], [(1,7),(3,9),(4,10),(6,12),(13,19),(14,20),(16,22),(17,23)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,20,19,14),(15,18,21,24),(16,23,22,17)]])
G:=TransitiveGroup(24,319);
►On 24 points - transitive group 24T395
Generators in S24
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 23)(2 18)(3 9)(4 14)(5 21)(6 12)(7 17)(8 24)(10 20)(11 15)(13 19)(16 22)
(1 17)(2 8)(3 13)(4 20)(5 11)(6 16)(7 23)(9 19)(10 14)(12 22)(15 21)(18 24)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 20 19 14)(15 18 21 24)(16 23 22 17)
G:=sub<Sym(24)| (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,23)(2,18)(3,9)(4,14)(5,21)(6,12)(7,17)(8,24)(10,20)(11,15)(13,19)(16,22), (1,17)(2,8)(3,13)(4,20)(5,11)(6,16)(7,23)(9,19)(10,14)(12,22)(15,21)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17)>;
G:=Group( (1,23)(2,24)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,23)(2,18)(3,9)(4,14)(5,21)(6,12)(7,17)(8,24)(10,20)(11,15)(13,19)(16,22), (1,17)(2,8)(3,13)(4,20)(5,11)(6,16)(7,23)(9,19)(10,14)(12,22)(15,21)(18,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,20,19,14)(15,18,21,24)(16,23,22,17) );
G=PermutationGroup([[(1,23),(2,24),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,23),(2,18),(3,9),(4,14),(5,21),(6,12),(7,17),(8,24),(10,20),(11,15),(13,19),(16,22)], [(1,17),(2,8),(3,13),(4,20),(5,11),(6,16),(7,23),(9,19),(10,14),(12,22),(15,21),(18,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,20,19,14),(15,18,21,24),(16,23,22,17)]])
G:=TransitiveGroup(24,395);
Matrix representation of C24.10D6 ►in GL5(𝔽13)
| 0 | 5 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 0 | 0 |
G:=sub<GL(5,GF(13))| [0,8,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[8,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,12,0,0],[0,8,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,12,0,0] >;
C24.10D6 in GAP, Magma, Sage, TeX
C_2^4._{10}D_6 % in TeX
G:=Group("C2^4.10D6"); // GroupNames label
G:=SmallGroup(192,1471);
// by ID
G=gap.SmallGroup(192,1471);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,64,254,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=b,f*a*f^-1=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,b*f=f*b,f*c*f^-1=e*d*e^-1=c*d=d*c,e*c*e^-1=d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations
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