p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2.6C25, C4.14C24, D4.9C23, C4○2- 1+4, C4○2+ 1+4, Q8.9C23, C22.4C24, C23.27C23, 2- 1+4⋊4C2, 2+ 1+4⋊5C2, D4○(C4○D4), Q8○(C4○D4), C4○D4⋊8C22, (C2×D4)⋊18C22, (C2×C4).47C23, (C2×Q8)⋊18C22, (C22×C4)⋊14C22, C4○D4○(C4○D4), (C2×C4○D4)⋊15C2, SmallGroup(64,266)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.C25
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f2=a, cbc=ebe=ab=ba, dcd=ac=ca, ad=da, ae=ea, af=fa, bd=db, bf=fb, ce=ec, cf=fc, de=ed, df=fd, ef=fe >
Subgroups: 465 in 405 conjugacy classes, 375 normal (4 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2.C25
Quotients: C1, C2, C22, C23, C24, C25, C2.C25
►On 16 points - transitive group 16T67
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 10)(2 11)(3 12)(4 9)(5 14)(6 15)(7 16)(8 13)
(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 15)(10 16)(11 13)(12 14)
(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)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14), (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( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,7)(2,8)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14), (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([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,10),(2,11),(3,12),(4,9),(5,14),(6,15),(7,16),(8,13)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,7),(2,8),(3,5),(4,6),(9,15),(10,16),(11,13),(12,14)], [(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(16,67);
C2.C25 is a maximal subgroup of
2+ 1+4.2C4 2+ 1+4⋊4C4 2- 1+4⋊5C4 C42.313C23 C42.12C23 C42.13C23 C23.7C24 C23.9C24 C23.10C24 2- 1+4⋊3C6 2+ 1+4.3C6 2- 1+4.C10
C2p.C25: C4.22C25 C8.C24 D8⋊C23 C4.C25 2+ 1+6 2- 1+6 C6.C25 D6.C24 ...
C2.C25 is a maximal quotient of
C22.14C25 C4×2+ 1+4 C4×2- 1+4 C22.38C25 C22.44C25 C22.47C25 C22.48C25 C22.49C25 C22.50C25 D4×C4○D4 C22.64C25 Q8×C4○D4 C22.69C25 C22.74C25 C22.76C25 C22.80C25 C22.82C25 C22.83C25 C22.84C25 C4⋊2+ 1+4 C4⋊2- 1+4 C22.90C25 C22.91C25 C22.93C25 C22.94C25 C22.95C25 C22.96C25 C22.99C25 C22.101C25 C22.102C25 C22.103C25 C22.104C25 C22.105C25 C22.106C25 C22.107C25 C22.110C25 C22.113C25 C22.118C25 C22.120C25 C22.122C25 C22.123C25 C22.124C25 C22.128C25 C22.129C25 C22.130C25 C22.131C25 C22.134C25 C22.135C25 C22.136C25 C22.140C25 C22.142C25 C22.143C25 C22.144C25 C22.146C25 C22.147C25 C22.148C25 C22.149C25 C22.150C25 C22.151C25 C22.152C25 C22.153C25 C22.154C25 C22.155C25 C22.156C25 C22.157C25
C4○D4⋊D2p: C22.77C25 C22.78C25 C22.89C25 C6.C25 D6.C24 D12.39C23 C10.C25 D20.37C23 ...
34 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2P | 4A | 4B | 4C | ··· | 4Q |
| order | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 |
| size | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2.C25 |
| kernel | C2.C25 | C2×C4○D4 | 2+ 1+4 | 2- 1+4 | C1 |
| # reps | 1 | 15 | 10 | 6 | 2 |
Matrix representation of C2.C25 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 1 |
| 4 | 0 | 3 | 2 |
| 2 | 2 | 0 | 1 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 1 | 0 |
| 0 | 0 | 2 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 3 | 4 | 0 |
| 2 | 0 | 3 | 0 |
| 0 | 0 | 1 | 1 |
| 4 | 0 | 3 | 0 |
| 1 | 1 | 2 | 0 |
| 4 | 3 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 3 | 0 | 1 |
| 0 | 2 | 1 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,4,2,0,0,0,2,0,0,3,0,0,1,2,1,4],[4,0,0,0,0,0,0,3,1,2,1,4,0,2,0,0],[2,0,4,1,0,0,0,1,3,1,3,2,0,1,0,0],[4,0,0,0,3,1,3,2,0,0,0,1,0,0,1,0],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2] >;
C2.C25 in GAP, Magma, Sage, TeX
C_2.C_2^5
% in TeX
G:=Group("C2.C2^5"); // GroupNames label
G:=SmallGroup(64,266);
// by ID
G=gap.SmallGroup(64,266);
# by ID
G:=PCGroup([6,-2,2,2,2,2,-2,409,332,963,88]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=1,f^2=a,c*b*c=e*b*e=a*b=b*a,d*c*d=a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*d=d*b,b*f=f*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations