p-group, metabelian, nilpotent (class 3), monomial
Aliases: C24.28D4, C24.2C23, C24⋊2(C2×C4), C22≀C2⋊1C4, C22⋊C4⋊31D4, (C2×D4).82D4, C22.11(C4×D4), (C22×C4).70D4, C22.D4⋊1C4, C23.136(C2×D4), C23⋊3D4.4C2, C22.11C24⋊5C2, C22.36C22≀C2, C23.9D4⋊10C2, C22.4(C4⋊D4), C23.70(C22×C4), C23.124(C4○D4), C23.16(C22⋊C4), (C22×D4).34C22, C2.52(C23.23D4), C22.11(C22.D4), (C2×C23⋊C4)⋊6C2, C22⋊C4⋊7(C2×C4), (C22×C4)⋊2(C2×C4), (C2×D4).91(C2×C4), (C2×C4).20(C22⋊C4), C22.51(C2×C22⋊C4), (C2×C22⋊C4).14C22, SmallGroup(128,645)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C24.28D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, eae-1=faf-1=acd, bc=cb, ebe-1=fbf-1=bd=db, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=bde-1 >
Subgroups: 492 in 190 conjugacy classes, 54 normal (20 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C22.D4, C22×D4, C23.9D4, C2×C23⋊C4, C2×C23⋊C4, C22.11C24, C23⋊3D4, C24.28D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C24.28D4
Character table of C24.28D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | -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 | -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 | 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 | -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 | 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 | 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 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | -i | i | -i | 1 | -1 | -1 | i | i | -i | 1 | -i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | i | -i | i | -i | i | -i | -i | -1 | -1 | 1 | i | -i | -i | -1 | i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | i | -i | i | i | -i | i | -i | 1 | 1 | 1 | -i | -i | i | -1 | i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | i | -i | i | -i | -i | -1 | 1 | -1 | -i | i | i | 1 | -i | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | i | -i | i | 1 | -1 | -1 | -i | -i | i | 1 | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | -i | i | -i | i | -i | i | i | -1 | -1 | 1 | -i | i | i | -1 | -i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -i | i | -i | -i | i | -i | i | 1 | 1 | 1 | i | i | -i | -1 | -i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | -i | i | -i | i | -i | i | i | -1 | 1 | -1 | i | -i | -i | 1 | i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 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 16T218
Generators in S16
(2 11)(3 13)(4 6)(5 12)(8 16)(9 14)
(2 16)(4 14)(6 9)(8 11)
(1 7)(2 11)(3 5)(4 9)(6 14)(8 16)(10 15)(12 13)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14 15 4)(2 13 16 3)(5 11 12 8)(6 10 9 7)
G:=sub<Sym(16)| (2,11)(3,13)(4,6)(5,12)(8,16)(9,14), (2,16)(4,14)(6,9)(8,11), (1,7)(2,11)(3,5)(4,9)(6,14)(8,16)(10,15)(12,13), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,15,4)(2,13,16,3)(5,11,12,8)(6,10,9,7)>;
G:=Group( (2,11)(3,13)(4,6)(5,12)(8,16)(9,14), (2,16)(4,14)(6,9)(8,11), (1,7)(2,11)(3,5)(4,9)(6,14)(8,16)(10,15)(12,13), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,15,4)(2,13,16,3)(5,11,12,8)(6,10,9,7) );
G=PermutationGroup([[(2,11),(3,13),(4,6),(5,12),(8,16),(9,14)], [(2,16),(4,14),(6,9),(8,11)], [(1,7),(2,11),(3,5),(4,9),(6,14),(8,16),(10,15),(12,13)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,15,4),(2,13,16,3),(5,11,12,8),(6,10,9,7)]])
G:=TransitiveGroup(16,218);
►On 16 points - transitive group 16T224
Generators in S16
(1 5)(2 6)(3 10)(4 11)(7 13)(8 14)(9 16)(12 15)
(1 5)(2 9)(3 7)(4 11)(6 16)(8 14)(10 13)(12 15)
(1 15)(3 13)(5 12)(7 10)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3 15 13)(2 9 16 6)(4 8 14 11)(5 10 12 7)
G:=sub<Sym(16)| (1,5)(2,6)(3,10)(4,11)(7,13)(8,14)(9,16)(12,15), (1,5)(2,9)(3,7)(4,11)(6,16)(8,14)(10,13)(12,15), (1,15)(3,13)(5,12)(7,10), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3,15,13)(2,9,16,6)(4,8,14,11)(5,10,12,7)>;
G:=Group( (1,5)(2,6)(3,10)(4,11)(7,13)(8,14)(9,16)(12,15), (1,5)(2,9)(3,7)(4,11)(6,16)(8,14)(10,13)(12,15), (1,15)(3,13)(5,12)(7,10), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3,15,13)(2,9,16,6)(4,8,14,11)(5,10,12,7) );
G=PermutationGroup([[(1,5),(2,6),(3,10),(4,11),(7,13),(8,14),(9,16),(12,15)], [(1,5),(2,9),(3,7),(4,11),(6,16),(8,14),(10,13),(12,15)], [(1,15),(3,13),(5,12),(7,10)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3,15,13),(2,9,16,6),(4,8,14,11),(5,10,12,7)]])
G:=TransitiveGroup(16,224);
►On 16 points - transitive group 16T268
Generators in S16
(1 10)(2 8)(3 5)(4 9)(6 14)(7 15)(11 16)(12 13)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)
(2 16)(4 14)(6 9)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3 15 13)(2 14 16 4)(5 10 12 7)(6 11 9 8)
G:=sub<Sym(16)| (1,10)(2,8)(3,5)(4,9)(6,14)(7,15)(11,16)(12,13), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15), (2,16)(4,14)(6,9)(8,11), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3,15,13)(2,14,16,4)(5,10,12,7)(6,11,9,8)>;
G:=Group( (1,10)(2,8)(3,5)(4,9)(6,14)(7,15)(11,16)(12,13), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15), (2,16)(4,14)(6,9)(8,11), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3,15,13)(2,14,16,4)(5,10,12,7)(6,11,9,8) );
G=PermutationGroup([[(1,10),(2,8),(3,5),(4,9),(6,14),(7,15),(11,16),(12,13)], [(1,3),(2,14),(4,16),(5,10),(6,8),(7,12),(9,11),(13,15)], [(2,16),(4,14),(6,9),(8,11)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3,15,13),(2,14,16,4),(5,10,12,7),(6,11,9,8)]])
G:=TransitiveGroup(16,268);
►On 16 points - transitive group 16T284
Generators in S16
(2 16)(3 13)(5 12)(6 9)
(2 16)(4 14)(6 9)(8 11)
(2 16)(4 14)(5 12)(7 10)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 15 9)(2 5 16 12)(3 11 13 8)(4 10 14 7)
G:=sub<Sym(16)| (2,16)(3,13)(5,12)(6,9), (2,16)(4,14)(6,9)(8,11), (2,16)(4,14)(5,12)(7,10), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,15,9)(2,5,16,12)(3,11,13,8)(4,10,14,7)>;
G:=Group( (2,16)(3,13)(5,12)(6,9), (2,16)(4,14)(6,9)(8,11), (2,16)(4,14)(5,12)(7,10), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,15,9)(2,5,16,12)(3,11,13,8)(4,10,14,7) );
G=PermutationGroup([[(2,16),(3,13),(5,12),(6,9)], [(2,16),(4,14),(6,9),(8,11)], [(2,16),(4,14),(5,12),(7,10)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6,15,9),(2,5,16,12),(3,11,13,8),(4,10,14,7)]])
G:=TransitiveGroup(16,284);
►On 16 points - transitive group 16T295
Generators in S16
(1 9)(2 10)(3 16)(4 15)(5 13)(6 14)(7 12)(8 11)
(1 8)(3 6)(9 11)(14 16)
(1 8)(4 5)(9 11)(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 13 8 15)(2 14 7 16)(3 12 6 10)(4 9 5 11)
G:=sub<Sym(16)| (1,9)(2,10)(3,16)(4,15)(5,13)(6,14)(7,12)(8,11), (1,8)(3,6)(9,11)(14,16), (1,8)(4,5)(9,11)(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), (1,13,8,15)(2,14,7,16)(3,12,6,10)(4,9,5,11)>;
G:=Group( (1,9)(2,10)(3,16)(4,15)(5,13)(6,14)(7,12)(8,11), (1,8)(3,6)(9,11)(14,16), (1,8)(4,5)(9,11)(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), (1,13,8,15)(2,14,7,16)(3,12,6,10)(4,9,5,11) );
G=PermutationGroup([[(1,9),(2,10),(3,16),(4,15),(5,13),(6,14),(7,12),(8,11)], [(1,8),(3,6),(9,11),(14,16)], [(1,8),(4,5),(9,11),(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)], [(1,13,8,15),(2,14,7,16),(3,12,6,10),(4,9,5,11)]])
G:=TransitiveGroup(16,295);
►On 16 points - transitive group 16T304
Generators in S16
(1 9)(2 10)(3 12)(4 11)(5 15)(6 14)(7 13)(8 16)
(1 6)(2 7)(3 5)(4 8)(9 14)(10 13)(11 16)(12 15)
(1 4)(6 8)(9 11)(14 16)
(1 4)(2 3)(5 7)(6 8)(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 14 4 16)(2 10 3 12)(5 13 7 15)(6 11 8 9)
G:=sub<Sym(16)| (1,9)(2,10)(3,12)(4,11)(5,15)(6,14)(7,13)(8,16), (1,6)(2,7)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15), (1,4)(6,8)(9,11)(14,16), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,4,16)(2,10,3,12)(5,13,7,15)(6,11,8,9)>;
G:=Group( (1,9)(2,10)(3,12)(4,11)(5,15)(6,14)(7,13)(8,16), (1,6)(2,7)(3,5)(4,8)(9,14)(10,13)(11,16)(12,15), (1,4)(6,8)(9,11)(14,16), (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,4,16)(2,10,3,12)(5,13,7,15)(6,11,8,9) );
G=PermutationGroup([[(1,9),(2,10),(3,12),(4,11),(5,15),(6,14),(7,13),(8,16)], [(1,6),(2,7),(3,5),(4,8),(9,14),(10,13),(11,16),(12,15)], [(1,4),(6,8),(9,11),(14,16)], [(1,4),(2,3),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,14,4,16),(2,10,3,12),(5,13,7,15),(6,11,8,9)]])
G:=TransitiveGroup(16,304);
►On 16 points - transitive group 16T318
Generators in S16
(1 2)(3 11)(4 10)(5 8)(6 13)(7 16)(9 12)(14 15)
(1 3)(2 11)(4 9)(5 7)(6 14)(8 16)(10 12)(13 15)
(1 3)(2 11)(4 9)(5 13)(6 8)(7 15)(10 12)(14 16)
(1 12)(2 9)(3 10)(4 11)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 15 12 5)(2 6 9 16)(3 7 10 13)(4 14 11 8)
G:=sub<Sym(16)| (1,2)(3,11)(4,10)(5,8)(6,13)(7,16)(9,12)(14,15), (1,3)(2,11)(4,9)(5,7)(6,14)(8,16)(10,12)(13,15), (1,3)(2,11)(4,9)(5,13)(6,8)(7,15)(10,12)(14,16), (1,12)(2,9)(3,10)(4,11)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,15,12,5)(2,6,9,16)(3,7,10,13)(4,14,11,8)>;
G:=Group( (1,2)(3,11)(4,10)(5,8)(6,13)(7,16)(9,12)(14,15), (1,3)(2,11)(4,9)(5,7)(6,14)(8,16)(10,12)(13,15), (1,3)(2,11)(4,9)(5,13)(6,8)(7,15)(10,12)(14,16), (1,12)(2,9)(3,10)(4,11)(5,15)(6,16)(7,13)(8,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,15,12,5)(2,6,9,16)(3,7,10,13)(4,14,11,8) );
G=PermutationGroup([[(1,2),(3,11),(4,10),(5,8),(6,13),(7,16),(9,12),(14,15)], [(1,3),(2,11),(4,9),(5,7),(6,14),(8,16),(10,12),(13,15)], [(1,3),(2,11),(4,9),(5,13),(6,8),(7,15),(10,12),(14,16)], [(1,12),(2,9),(3,10),(4,11),(5,15),(6,16),(7,13),(8,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,15,12,5),(2,6,9,16),(3,7,10,13),(4,14,11,8)]])
G:=TransitiveGroup(16,318);
►On 16 points - transitive group 16T326
Generators in S16
(1 9)(2 13)(3 14)(4 12)(5 16)(6 10)(7 11)(8 15)
(2 8)(4 6)(10 12)(13 15)
(1 3)(2 6)(4 8)(5 7)(9 14)(10 13)(11 16)(12 15)
(1 7)(2 8)(3 5)(4 6)(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 12 7 10)(2 11 8 9)(3 15 5 13)(4 14 6 16)
G:=sub<Sym(16)| (1,9)(2,13)(3,14)(4,12)(5,16)(6,10)(7,11)(8,15), (2,8)(4,6)(10,12)(13,15), (1,3)(2,6)(4,8)(5,7)(9,14)(10,13)(11,16)(12,15), (1,7)(2,8)(3,5)(4,6)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,12,7,10)(2,11,8,9)(3,15,5,13)(4,14,6,16)>;
G:=Group( (1,9)(2,13)(3,14)(4,12)(5,16)(6,10)(7,11)(8,15), (2,8)(4,6)(10,12)(13,15), (1,3)(2,6)(4,8)(5,7)(9,14)(10,13)(11,16)(12,15), (1,7)(2,8)(3,5)(4,6)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,12,7,10)(2,11,8,9)(3,15,5,13)(4,14,6,16) );
G=PermutationGroup([[(1,9),(2,13),(3,14),(4,12),(5,16),(6,10),(7,11),(8,15)], [(2,8),(4,6),(10,12),(13,15)], [(1,3),(2,6),(4,8),(5,7),(9,14),(10,13),(11,16),(12,15)], [(1,7),(2,8),(3,5),(4,6),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,12,7,10),(2,11,8,9),(3,15,5,13),(4,14,6,16)]])
G:=TransitiveGroup(16,326);
Matrix representation of C24.28D4 ►in GL8(ℤ)
| 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 |
| 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 | 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 |
| 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 | -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 | -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 | -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 | -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 | 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 |
G:=sub<GL(8,Integers())| [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],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[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,-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],[0,0,0,0,0,1,0,0,0,0,0,0,-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],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-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] >;
C24.28D4 in GAP, Magma, Sage, TeX
C_2^4._{28}D_4 % in TeX
G:=Group("C2^4.28D4"); // GroupNames label
G:=SmallGroup(128,645);
// by ID
G=gap.SmallGroup(128,645);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,521,2804,1411,2028,1027]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=f*a*f^-1=a*c*d,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*d*e^-1>;
// generators/relations
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