Copied to
clipboard

G = C2.C25order 64 = 26

6th central stem extension by C2 of C25

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2.6C25, C4.14C24, D4.9C23, C42- 1+4, C42+ 1+4, Q8.9C23, C22.4C24, C23.27C23, 2- 1+44C2, 2+ 1+45C2, D4(C4○D4), Q8(C4○D4), C4○D48C22, (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

C1C2 — C2.C25
C1C2C4C2×C4C22×C4C2×C4○D4 — C2.C25
C1C2 — C2.C25
C1C4 — C2.C25
C1C2 — C2.C25

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 [×15], C4, C4 [×15], C22 [×15], C22 [×15], C2×C4 [×60], D4 [×60], Q8 [×20], C23 [×15], C22×C4 [×15], C2×D4 [×45], C2×Q8 [×15], C4○D4 [×80], C2×C4○D4 [×15], 2+ 1+4 [×10], 2- 1+4 [×6], C2.C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], C25, C2.C25

Permutation representations of C2.C25
On 16 points - transitive group 16T67
Generators in S16
(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+44C4  2- 1+45C4  C42.313C23  C42.12C23  C42.13C23  C23.7C24  C23.9C24  C23.10C24  2- 1+43C6  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 2A2B···2P4A4B4C···4Q
order122···2444···4
size112···2112···2

34 irreducible representations

dim11114
type++++
imageC1C2C2C2C2.C25
kernelC2.C25C2×C4○D42+ 1+42- 1+4C1
# reps1151062

Matrix representation of C2.C25 in GL4(𝔽5) generated by

4000
0400
0040
0004
,
1001
4032
2201
0004
,
4010
0022
0010
0340
,
2030
0011
4030
1120
,
4300
0100
0301
0210
,
2000
0200
0020
0002
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

׿
×
𝔽