p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C24⋊5D4, C24⋊2C23, C23.8C24, 2+ (1+4).14C22, (C2×D4)⋊26D4, (C22×C4)⋊7D4, C2≀C22⋊4C2, C23⋊3D4⋊3C2, C22⋊C4⋊3C23, C23⋊C4⋊8C22, (C22×C4)⋊2C23, C23.27(C2×D4), C22≀C2⋊1C22, C22.6C22≀C2, (C2×D4).42C23, C23.7D4⋊4C2, (C22×D4)⋊21C22, (C2×2+ (1+4))⋊6C2, C22.42(C22×D4), C22.D4⋊1C22, (C2×C4).28(C2×D4), (C2×C23⋊C4)⋊18C2, C2.63(C2×C22≀C2), (C2×C22⋊C4)⋊38C22, SmallGroup(128,1758)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 924 in 400 conjugacy classes, 106 normal (10 characteristic)
C1, C2, C2 [×14], C4 [×13], C22, C22 [×6], C22 [×39], C2×C4 [×6], C2×C4 [×30], D4 [×46], Q8 [×4], C23, C23 [×12], C23 [×20], C22⋊C4 [×6], C22⋊C4 [×15], C4⋊C4 [×6], C22×C4, C22×C4 [×3], C22×C4 [×6], C2×D4 [×12], C2×D4 [×51], C2×Q8, C4○D4 [×24], C24 [×2], C24 [×3], C24, C23⋊C4 [×12], C2×C22⋊C4 [×3], C22≀C2 [×6], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×6], C22.D4 [×3], C22×D4, C22×D4 [×3], C22×D4 [×3], C2×C4○D4 [×3], 2+ (1+4) [×4], 2+ (1+4) [×6], C2×C23⋊C4 [×3], C2≀C22 [×4], C23.7D4 [×4], C23⋊3D4 [×3], C2×2+ (1+4), C24⋊C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C24⋊C23
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, eae=ab=ba, faf=ac=ca, gag=ad=da, bc=cb, fbf=bd=db, be=eb, bg=gb, ece=cd=dc, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >
►On 16 points - transitive group 16T241
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(3 10)(4 9)(7 11)(8 12)
(3 10)(4 9)(5 15)(6 16)
(1 13)(2 14)(3 10)(4 9)(5 15)(6 16)(7 11)(8 12)
(1 5)(2 6)(3 11)(4 8)(7 10)(9 12)(13 15)(14 16)
(1 11)(2 12)(3 5)(4 16)(6 9)(7 13)(8 14)(10 15)
(2 14)(4 9)(6 16)(8 12)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (3,10)(4,9)(7,11)(8,12), (3,10)(4,9)(5,15)(6,16), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12), (1,5)(2,6)(3,11)(4,8)(7,10)(9,12)(13,15)(14,16), (1,11)(2,12)(3,5)(4,16)(6,9)(7,13)(8,14)(10,15), (2,14)(4,9)(6,16)(8,12)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (3,10)(4,9)(7,11)(8,12), (3,10)(4,9)(5,15)(6,16), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12), (1,5)(2,6)(3,11)(4,8)(7,10)(9,12)(13,15)(14,16), (1,11)(2,12)(3,5)(4,16)(6,9)(7,13)(8,14)(10,15), (2,14)(4,9)(6,16)(8,12) );
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(3,10),(4,9),(7,11),(8,12)], [(3,10),(4,9),(5,15),(6,16)], [(1,13),(2,14),(3,10),(4,9),(5,15),(6,16),(7,11),(8,12)], [(1,5),(2,6),(3,11),(4,8),(7,10),(9,12),(13,15),(14,16)], [(1,11),(2,12),(3,5),(4,16),(6,9),(7,13),(8,14),(10,15)], [(2,14),(4,9),(6,16),(8,12)])
G:=TransitiveGroup(16,241);
►On 16 points - transitive group 16T277
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 9)(2 10)(3 16)(4 15)(5 11)(6 12)(7 13)(8 14)
(1 14)(2 13)(3 5)(4 6)(7 10)(8 9)(11 16)(12 15)
(1 11)(2 12)(3 8)(4 7)(5 9)(6 10)(13 15)(14 16)
(1 9)(3 14)(4 7)(5 11)(8 16)(13 15)
(1 16)(2 12)(3 5)(8 9)(11 14)(13 15)
(2 12)(4 7)(6 10)(13 15)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,9)(2,10)(3,16)(4,15)(5,11)(6,12)(7,13)(8,14), (1,14)(2,13)(3,5)(4,6)(7,10)(8,9)(11,16)(12,15), (1,11)(2,12)(3,8)(4,7)(5,9)(6,10)(13,15)(14,16), (1,9)(3,14)(4,7)(5,11)(8,16)(13,15), (1,16)(2,12)(3,5)(8,9)(11,14)(13,15), (2,12)(4,7)(6,10)(13,15)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,9)(2,10)(3,16)(4,15)(5,11)(6,12)(7,13)(8,14), (1,14)(2,13)(3,5)(4,6)(7,10)(8,9)(11,16)(12,15), (1,11)(2,12)(3,8)(4,7)(5,9)(6,10)(13,15)(14,16), (1,9)(3,14)(4,7)(5,11)(8,16)(13,15), (1,16)(2,12)(3,5)(8,9)(11,14)(13,15), (2,12)(4,7)(6,10)(13,15) );
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,9),(2,10),(3,16),(4,15),(5,11),(6,12),(7,13),(8,14)], [(1,14),(2,13),(3,5),(4,6),(7,10),(8,9),(11,16),(12,15)], [(1,11),(2,12),(3,8),(4,7),(5,9),(6,10),(13,15),(14,16)], [(1,9),(3,14),(4,7),(5,11),(8,16),(13,15)], [(1,16),(2,12),(3,5),(8,9),(11,14),(13,15)], [(2,12),(4,7),(6,10),(13,15)])
G:=TransitiveGroup(16,277);
►On 16 points - transitive group 16T301
Generators in S16
(9 10)(11 12)(13 14)(15 16)
(1 3)(2 6)(9 10)(13 14)
(2 6)(4 8)(13 14)(15 16)
(1 3)(2 6)(4 8)(5 7)(9 10)(11 12)(13 14)(15 16)
(1 13)(2 9)(3 14)(4 7)(5 8)(6 10)(11 15)(12 16)
(1 7)(2 15)(3 5)(4 13)(6 16)(8 14)(9 11)(10 12)
(1 9)(2 13)(3 10)(4 15)(5 12)(6 14)(7 11)(8 16)
G:=sub<Sym(16)| (9,10)(11,12)(13,14)(15,16), (1,3)(2,6)(9,10)(13,14), (2,6)(4,8)(13,14)(15,16), (1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,14)(15,16), (1,13)(2,9)(3,14)(4,7)(5,8)(6,10)(11,15)(12,16), (1,7)(2,15)(3,5)(4,13)(6,16)(8,14)(9,11)(10,12), (1,9)(2,13)(3,10)(4,15)(5,12)(6,14)(7,11)(8,16)>;
G:=Group( (9,10)(11,12)(13,14)(15,16), (1,3)(2,6)(9,10)(13,14), (2,6)(4,8)(13,14)(15,16), (1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,14)(15,16), (1,13)(2,9)(3,14)(4,7)(5,8)(6,10)(11,15)(12,16), (1,7)(2,15)(3,5)(4,13)(6,16)(8,14)(9,11)(10,12), (1,9)(2,13)(3,10)(4,15)(5,12)(6,14)(7,11)(8,16) );
G=PermutationGroup([(9,10),(11,12),(13,14),(15,16)], [(1,3),(2,6),(9,10),(13,14)], [(2,6),(4,8),(13,14),(15,16)], [(1,3),(2,6),(4,8),(5,7),(9,10),(11,12),(13,14),(15,16)], [(1,13),(2,9),(3,14),(4,7),(5,8),(6,10),(11,15),(12,16)], [(1,7),(2,15),(3,5),(4,13),(6,16),(8,14),(9,11),(10,12)], [(1,9),(2,13),(3,10),(4,15),(5,12),(6,14),(7,11),(8,16)])
G:=TransitiveGroup(16,301);
►On 16 points - transitive group 16T309
Generators in S16
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(5 8)(6 7)(9 14)(10 13)
(1 4)(2 3)(5 7)(6 8)(9 10)(11 16)(12 15)(13 14)
(1 2)(3 4)(5 8)(6 7)(9 14)(10 13)(11 12)(15 16)
(3 4)(5 8)(10 13)(15 16)
(1 14)(2 9)(3 10)(4 13)(5 15)(6 11)(7 12)(8 16)
(1 11)(2 12)(3 15)(4 16)(5 10)(6 14)(7 9)(8 13)
G:=sub<Sym(16)| (5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (5,8)(6,7)(9,14)(10,13), (1,4)(2,3)(5,7)(6,8)(9,10)(11,16)(12,15)(13,14), (1,2)(3,4)(5,8)(6,7)(9,14)(10,13)(11,12)(15,16), (3,4)(5,8)(10,13)(15,16), (1,14)(2,9)(3,10)(4,13)(5,15)(6,11)(7,12)(8,16), (1,11)(2,12)(3,15)(4,16)(5,10)(6,14)(7,9)(8,13)>;
G:=Group( (5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (5,8)(6,7)(9,14)(10,13), (1,4)(2,3)(5,7)(6,8)(9,10)(11,16)(12,15)(13,14), (1,2)(3,4)(5,8)(6,7)(9,14)(10,13)(11,12)(15,16), (3,4)(5,8)(10,13)(15,16), (1,14)(2,9)(3,10)(4,13)(5,15)(6,11)(7,12)(8,16), (1,11)(2,12)(3,15)(4,16)(5,10)(6,14)(7,9)(8,13) );
G=PermutationGroup([(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(5,8),(6,7),(9,14),(10,13)], [(1,4),(2,3),(5,7),(6,8),(9,10),(11,16),(12,15),(13,14)], [(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12),(15,16)], [(3,4),(5,8),(10,13),(15,16)], [(1,14),(2,9),(3,10),(4,13),(5,15),(6,11),(7,12),(8,16)], [(1,11),(2,12),(3,15),(4,16),(5,10),(6,14),(7,9),(8,13)])
G:=TransitiveGroup(16,309);
►On 16 points - transitive group 16T320
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 9)(2 10)(3 13)(4 14)(5 7)(6 8)(11 16)(12 15)
(1 5)(2 6)(3 13)(4 14)(7 9)(8 10)(11 16)(12 15)
(1 7)(2 8)(3 15)(4 16)(5 9)(6 10)(11 14)(12 13)
(1 16)(2 12)(3 6)(4 7)(5 14)(8 13)(9 11)(10 15)
(1 12)(2 16)(3 9)(4 8)(5 15)(6 11)(7 13)(10 14)
(1 5)(2 10)(3 13)(4 11)(6 8)(7 9)(12 15)(14 16)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,9)(2,10)(3,13)(4,14)(5,7)(6,8)(11,16)(12,15), (1,5)(2,6)(3,13)(4,14)(7,9)(8,10)(11,16)(12,15), (1,7)(2,8)(3,15)(4,16)(5,9)(6,10)(11,14)(12,13), (1,16)(2,12)(3,6)(4,7)(5,14)(8,13)(9,11)(10,15), (1,12)(2,16)(3,9)(4,8)(5,15)(6,11)(7,13)(10,14), (1,5)(2,10)(3,13)(4,11)(6,8)(7,9)(12,15)(14,16)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,9)(2,10)(3,13)(4,14)(5,7)(6,8)(11,16)(12,15), (1,5)(2,6)(3,13)(4,14)(7,9)(8,10)(11,16)(12,15), (1,7)(2,8)(3,15)(4,16)(5,9)(6,10)(11,14)(12,13), (1,16)(2,12)(3,6)(4,7)(5,14)(8,13)(9,11)(10,15), (1,12)(2,16)(3,9)(4,8)(5,15)(6,11)(7,13)(10,14), (1,5)(2,10)(3,13)(4,11)(6,8)(7,9)(12,15)(14,16) );
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,9),(2,10),(3,13),(4,14),(5,7),(6,8),(11,16),(12,15)], [(1,5),(2,6),(3,13),(4,14),(7,9),(8,10),(11,16),(12,15)], [(1,7),(2,8),(3,15),(4,16),(5,9),(6,10),(11,14),(12,13)], [(1,16),(2,12),(3,6),(4,7),(5,14),(8,13),(9,11),(10,15)], [(1,12),(2,16),(3,9),(4,8),(5,15),(6,11),(7,13),(10,14)], [(1,5),(2,10),(3,13),(4,11),(6,8),(7,9),(12,15),(14,16)])
G:=TransitiveGroup(16,320);
►On 16 points - transitive group 16T329
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 11)(2 12)(3 15)(4 16)(5 10)(6 9)(7 13)(8 14)
(1 11)(2 12)(3 5)(4 6)(7 13)(8 14)(9 16)(10 15)
(1 13)(2 14)(3 10)(4 9)(5 15)(6 16)(7 11)(8 12)
(1 5)(2 9)(3 7)(4 14)(6 12)(8 16)(10 11)(13 15)
(1 5)(2 4)(3 11)(6 12)(7 10)(8 16)(9 14)(13 15)
(2 14)(4 9)(6 16)(8 12)
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,11)(2,12)(3,15)(4,16)(5,10)(6,9)(7,13)(8,14), (1,11)(2,12)(3,5)(4,6)(7,13)(8,14)(9,16)(10,15), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12), (1,5)(2,9)(3,7)(4,14)(6,12)(8,16)(10,11)(13,15), (1,5)(2,4)(3,11)(6,12)(7,10)(8,16)(9,14)(13,15), (2,14)(4,9)(6,16)(8,12)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,11)(2,12)(3,15)(4,16)(5,10)(6,9)(7,13)(8,14), (1,11)(2,12)(3,5)(4,6)(7,13)(8,14)(9,16)(10,15), (1,13)(2,14)(3,10)(4,9)(5,15)(6,16)(7,11)(8,12), (1,5)(2,9)(3,7)(4,14)(6,12)(8,16)(10,11)(13,15), (1,5)(2,4)(3,11)(6,12)(7,10)(8,16)(9,14)(13,15), (2,14)(4,9)(6,16)(8,12) );
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,11),(2,12),(3,15),(4,16),(5,10),(6,9),(7,13),(8,14)], [(1,11),(2,12),(3,5),(4,6),(7,13),(8,14),(9,16),(10,15)], [(1,13),(2,14),(3,10),(4,9),(5,15),(6,16),(7,11),(8,12)], [(1,5),(2,9),(3,7),(4,14),(6,12),(8,16),(10,11),(13,15)], [(1,5),(2,4),(3,11),(6,12),(7,10),(8,16),(9,14),(13,15)], [(2,14),(4,9),(6,16),(8,12)])
G:=TransitiveGroup(16,329);
Matrix representation ►G ⊆ GL8(ℤ)
| 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 | 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 | 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 | 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 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 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 |
| 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 | 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 |
G:=sub<GL(8,Integers())| [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],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[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,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-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] >;
Character table of C24⋊C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 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 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | -2 | 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 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ26 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ27 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ28 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 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 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes C_2^3
% in TeX
G:=Group("C2^4:C2^3"); // GroupNames label
G:=SmallGroup(128,1758);
// by ID
G=gap.SmallGroup(128,1758);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,718,2028]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,e*a*e=a*b=b*a,f*a*f=a*c=c*a,g*a*g=a*d=d*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,b*g=g*b,e*c*e=c*d=d*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations
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