p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.7C24, C24.471C23, 2+ 1+4.13C22, C4○C2≀C22, (C2×D4)⋊25D4, (C2×Q8)⋊20D4, (C22×C4)⋊6D4, C2≀C22⋊6C2, C4.88C22≀C2, C23⋊C4⋊7C22, C23.26(C2×D4), C2.C25⋊3C2, (C23×C4)⋊26C22, (C2×D4).41C23, C4○(C23.7D4), C23.7D4⋊6C2, C22⋊C4.1C23, C22≀C2⋊24C22, C22.19C24⋊5C2, C22.41(C22×D4), C42⋊C2⋊12C22, (C22×C4).616C23, C23.C23⋊18C2, C22.D4⋊29C22, (C2×C4).27(C2×D4), C2.62(C2×C22≀C2), (C2×C4○D4).111C22, SmallGroup(128,1757)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.7C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=c, ab=ba, faf=ac=ca, ede=ad=da, ae=ea, ag=ga, ebe=bc=cb, fdf=bd=db, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 772 in 383 conjugacy classes, 106 normal (10 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C23⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, C23.C23, C2≀C22, C23.7D4, C22.19C24, C2.C25, C23.7C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2, C23.7C24
►On 16 points - transitive group 16T223
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)
(1 14)(2 15)(3 16)(4 13)(5 9)(6 10)(7 11)(8 12)
(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)
G:=sub<Sym(16)| (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12), (1,14)(2,15)(3,16)(4,13)(5,9)(6,10)(7,11)(8,12), (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)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12), (1,14)(2,15)(3,16)(4,13)(5,9)(6,10)(7,11)(8,12), (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) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13)], [(1,11),(2,12),(3,9),(4,10),(5,14),(6,15),(7,16),(8,13)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,11),(10,12)], [(1,14),(2,15),(3,16),(4,13),(5,9),(6,10),(7,11),(8,12)], [(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)]])
G:=TransitiveGroup(16,223);
►On 16 points - transitive group 16T270
(5 7)(6 8)(13 15)(14 16)
(9 11)(10 12)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (5,7)(6,8)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,7)(2,8)(3,5)(4,6)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (5,7)(6,8)(13,15)(14,16), (9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,7)(2,8)(3,5)(4,6)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(5,7),(6,8),(13,15),(14,16)], [(9,11),(10,12),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,7),(2,8),(3,5),(4,6),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,270);
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2L | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | ··· | 4S |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C23.7C24 |
| kernel | C23.7C24 | C23.C23 | C2≀C22 | C23.7D4 | C22.19C24 | C2.C25 | C22×C4 | C2×D4 | C2×Q8 | C1 |
| # reps | 1 | 3 | 4 | 4 | 3 | 1 | 6 | 3 | 3 | 4 |
Matrix representation of C23.7C24 ►in GL4(𝔽5) generated by
| 4 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 4 | 0 | 1 |
| 1 | 4 | 1 | 0 |
| 1 | 3 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 4 | 0 | 3 | 0 |
| 4 | 0 | 4 | 4 |
| 0 | 0 | 1 | 0 |
| 1 | 4 | 1 | 0 |
| 4 | 0 | 3 | 0 |
| 0 | 0 | 4 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
G:=sub<GL(4,GF(5))| [4,0,1,1,2,1,4,4,0,0,0,1,0,0,1,0],[1,0,0,0,3,4,1,1,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[4,4,0,1,0,0,0,4,3,4,1,1,0,4,0,0],[4,0,0,0,0,0,0,1,3,4,1,1,0,1,0,0],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;
C23.7C24 in GAP, Magma, Sage, TeX
C_2^3._7C_2^4
% in TeX
G:=Group("C2^3.7C2^4"); // GroupNames label
G:=SmallGroup(128,1757);
// by ID
G=gap.SmallGroup(128,1757);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,248,718,2028]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=c,a*b=b*a,f*a*f=a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,a*g=g*a,e*b*e=b*c=c*b,f*d*f=b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations