p-group, metabelian, nilpotent (class 4), monomial
Aliases: C24.39D4, C4⋊1D4⋊5C4, C42⋊2(C2×C4), C4.4D4⋊5C4, C42⋊3C4⋊3C2, C42⋊C2⋊9C4, C42⋊C4⋊3C2, (C2×D4).132D4, C23.10(C2×D4), (C2×D4).21C23, C4⋊1D4.53C22, C23⋊C4.13C22, C22.9(C23⋊C4), C23.24(C22⋊C4), C22.29C24.7C2, C4.4D4.14C22, (C22×D4).103C22, (C2×C4○D4)⋊8C4, (C2×D4)⋊4(C2×C4), (C2×Q8)⋊4(C2×C4), (C2×C23⋊C4)⋊14C2, C2.39(C2×C23⋊C4), (C2×C4).96(C22×C4), (C22×C4).31(C2×C4), (C2×C4).51(C22⋊C4), C22.63(C2×C22⋊C4), SmallGroup(128,859)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.39D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=b, ab=ba, ac=ca, eae-1=ad=da, af=fa, ebe-1=bc=cb, bd=db, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be-1 >
Subgroups: 428 in 143 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C23⋊C4, C23⋊C4, C2×C22⋊C4, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C42⋊C4, C42⋊3C4, C2×C23⋊C4, C22.29C24, C24.39D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C24.39D4
Character table of C24.39D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -1 | 1 | i | i | -i | i | -i | -1 | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | i | -i | -1 | -1 | -i | i | i | i | -i | 1 | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | -i | 1 | -1 | i | i | i | -i | i | 1 | -i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | 1 | 1 | -i | i | -i | -i | i | -1 | -i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | i | -1 | -1 | i | -i | -i | -i | i | 1 | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -1 | 1 | -i | -i | i | -i | i | -1 | i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | 1 | 1 | i | -i | i | i | -i | -1 | i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | i | i | 1 | -1 | -i | -i | -i | i | -i | 1 | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ22 | 4 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 16 points - transitive group 16T234
Generators in S16
(1 8)(2 11)(3 6)(4 9)(5 15)(7 13)(10 14)(12 16)
(1 8)(2 5)(3 12)(4 9)(6 16)(7 13)(10 14)(11 15)
(2 15)(4 13)(5 11)(7 9)
(1 14)(2 15)(3 16)(4 13)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 8 4)(2 3 5 12)(6 15 16 11)(7 10 13 14)
G:=sub<Sym(16)| (1,8)(2,11)(3,6)(4,9)(5,15)(7,13)(10,14)(12,16), (1,8)(2,5)(3,12)(4,9)(6,16)(7,13)(10,14)(11,15), (2,15)(4,13)(5,11)(7,9), (1,14)(2,15)(3,16)(4,13)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,8,4)(2,3,5,12)(6,15,16,11)(7,10,13,14)>;
G:=Group( (1,8)(2,11)(3,6)(4,9)(5,15)(7,13)(10,14)(12,16), (1,8)(2,5)(3,12)(4,9)(6,16)(7,13)(10,14)(11,15), (2,15)(4,13)(5,11)(7,9), (1,14)(2,15)(3,16)(4,13)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,8,4)(2,3,5,12)(6,15,16,11)(7,10,13,14) );
G=PermutationGroup([[(1,8),(2,11),(3,6),(4,9),(5,15),(7,13),(10,14),(12,16)], [(1,8),(2,5),(3,12),(4,9),(6,16),(7,13),(10,14),(11,15)], [(2,15),(4,13),(5,11),(7,9)], [(1,14),(2,15),(3,16),(4,13),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9,8,4),(2,3,5,12),(6,15,16,11),(7,10,13,14)]])
G:=TransitiveGroup(16,234);
►On 16 points - transitive group 16T250
Generators in S16
(1 3)(2 5)(4 7)(6 8)(9 14)(10 13)(11 16)(12 15)
(1 10)(2 9)(3 13)(4 16)(5 14)(6 15)(7 11)(8 12)
(1 8)(3 6)(10 12)(13 15)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 2 10 9)(3 5 13 14)(4 15 16 6)(7 12 11 8)
G:=sub<Sym(16)| (1,3)(2,5)(4,7)(6,8)(9,14)(10,13)(11,16)(12,15), (1,10)(2,9)(3,13)(4,16)(5,14)(6,15)(7,11)(8,12), (1,8)(3,6)(10,12)(13,15), (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2,10,9)(3,5,13,14)(4,15,16,6)(7,12,11,8)>;
G:=Group( (1,3)(2,5)(4,7)(6,8)(9,14)(10,13)(11,16)(12,15), (1,10)(2,9)(3,13)(4,16)(5,14)(6,15)(7,11)(8,12), (1,8)(3,6)(10,12)(13,15), (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,2,10,9)(3,5,13,14)(4,15,16,6)(7,12,11,8) );
G=PermutationGroup([[(1,3),(2,5),(4,7),(6,8),(9,14),(10,13),(11,16),(12,15)], [(1,10),(2,9),(3,13),(4,16),(5,14),(6,15),(7,11),(8,12)], [(1,8),(3,6),(10,12),(13,15)], [(1,8),(2,7),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,2,10,9),(3,5,13,14),(4,15,16,6),(7,12,11,8)]])
G:=TransitiveGroup(16,250);
►On 16 points - transitive group 16T297
Generators in S16
(1 4)(2 3)(5 7)(6 13)(8 15)(9 10)(11 12)(14 16)
(1 11)(2 9)(3 10)(4 12)(5 13)(6 7)(8 14)(15 16)
(1 3)(2 4)(5 7)(6 13)(8 15)(9 12)(10 11)(14 16)
(1 2)(3 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)
(3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16 11 15)(2 7 9 6)(3 5 10 13)(4 14 12 8)
G:=sub<Sym(16)| (1,4)(2,3)(5,7)(6,13)(8,15)(9,10)(11,12)(14,16), (1,11)(2,9)(3,10)(4,12)(5,13)(6,7)(8,14)(15,16), (1,3)(2,4)(5,7)(6,13)(8,15)(9,12)(10,11)(14,16), (1,2)(3,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,11,15)(2,7,9,6)(3,5,10,13)(4,14,12,8)>;
G:=Group( (1,4)(2,3)(5,7)(6,13)(8,15)(9,10)(11,12)(14,16), (1,11)(2,9)(3,10)(4,12)(5,13)(6,7)(8,14)(15,16), (1,3)(2,4)(5,7)(6,13)(8,15)(9,12)(10,11)(14,16), (1,2)(3,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16,11,15)(2,7,9,6)(3,5,10,13)(4,14,12,8) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,13),(8,15),(9,10),(11,12),(14,16)], [(1,11),(2,9),(3,10),(4,12),(5,13),(6,7),(8,14),(15,16)], [(1,3),(2,4),(5,7),(6,13),(8,15),(9,12),(10,11),(14,16)], [(1,2),(3,4),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12)], [(3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16,11,15),(2,7,9,6),(3,5,10,13),(4,14,12,8)]])
G:=TransitiveGroup(16,297);
►On 16 points - transitive group 16T310
Generators in S16
(1 6)(3 8)(10 12)(13 15)
(1 12)(2 9)(3 15)(4 14)(5 11)(6 10)(7 16)(8 13)
(2 5)(3 8)(9 11)(13 15)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 15 12 3)(2 4 9 14)(5 7 11 16)(6 13 10 8)
G:=sub<Sym(16)| (1,6)(3,8)(10,12)(13,15), (1,12)(2,9)(3,15)(4,14)(5,11)(6,10)(7,16)(8,13), (2,5)(3,8)(9,11)(13,15), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,15,12,3)(2,4,9,14)(5,7,11,16)(6,13,10,8)>;
G:=Group( (1,6)(3,8)(10,12)(13,15), (1,12)(2,9)(3,15)(4,14)(5,11)(6,10)(7,16)(8,13), (2,5)(3,8)(9,11)(13,15), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,15,12,3)(2,4,9,14)(5,7,11,16)(6,13,10,8) );
G=PermutationGroup([[(1,6),(3,8),(10,12),(13,15)], [(1,12),(2,9),(3,15),(4,14),(5,11),(6,10),(7,16),(8,13)], [(2,5),(3,8),(9,11),(13,15)], [(1,6),(2,5),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,15,12,3),(2,4,9,14),(5,7,11,16),(6,13,10,8)]])
G:=TransitiveGroup(16,310);
Matrix representation of C24.39D4 ►in GL8(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0] >;
C24.39D4 in GAP, Magma, Sage, TeX
C_2^4._{39}D_4 % in TeX
G:=Group("C2^4.39D4"); // GroupNames label
G:=SmallGroup(128,859);
// by ID
G=gap.SmallGroup(128,859);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,1018,248,1971,375,4037]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^-1>;
// generators/relations
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