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G = C25.C4  order 128 = 27

4th non-split extension by C25 of C4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C25.4C4, C4.49C22≀C2, (C2×D4).261D4, (D4×C23).3C2, C24.113(C2×C4), (C22×D4).27C4, (C22×C4).259D4, C24.4C4⋊23C2, C2.8(C24⋊3C4), C22⋊2(C4.D4), C23.29(C22⋊C4), (C2×M4(2))⋊36C22, (C22×C4).653C23, (C23×C4).228C22, C23.182(C22×C4), (C22×D4).450C22, (C2×C4.D4)⋊13C2, (C2×C4).1308(C2×D4), C2.24(C2×C4.D4), (C22×C4).258(C2×C4), C22.30(C2×C22⋊C4), (C2×C4).118(C22⋊C4), SmallGroup(128,515)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C25.C4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C25.C4
Lower central
C1 — C2 — C23 — C25.C4
Upper central
C1 — C22 — C23×C4 — C25.C4
Jennings
C1 — C2 — C2 — C22×C4 — C25.C4

Generators and relations for C25.C4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, faf-1=ace, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1012 in 410 conjugacy classes, 72 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C22⋊C8, C4.D4, C2×M4(2), C23×C4, C22×D4, C22×D4, C25, C24.4C4, C2×C4.D4, D4×C23, C25.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C22≀C2, C24⋊3C4, C2×C4.D4, C25.C4

Permutation representations of C25.C4
►On 16 points - transitive group 16T253
Generators in S16
(1 9)(2 8)(3 15)(4 6)(5 13)(7 11)(10 12)(14 16)

(1 5)(2 16)(3 7)(4 10)(6 12)(8 14)(9 13)(11 15)

(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)

(1 5)(2 6)(3 7)(4 8)(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)| (1,9)(2,8)(3,15)(4,6)(5,13)(7,11)(10,12)(14,16), (1,5)(2,16)(3,7)(4,10)(6,12)(8,14)(9,13)(11,15), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,5)(2,6)(3,7)(4,8)(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( (1,9)(2,8)(3,15)(4,6)(5,13)(7,11)(10,12)(14,16), (1,5)(2,16)(3,7)(4,10)(6,12)(8,14)(9,13)(11,15), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,5)(2,6)(3,7)(4,8)(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([[(1,9),(2,8),(3,15),(4,6),(5,13),(7,11),(10,12),(14,16)], [(1,5),(2,16),(3,7),(4,10),(6,12),(8,14),(9,13),(11,15)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10)], [(1,5),(2,6),(3,7),(4,8),(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(16,253);
 

32 conjugacy classes

class 1 2A2B2C2D···2I2J···2Q4A4B4C4D4E4F8A···8H
order12222···22···24444448···8
size11112···24···42222448···8

32 irreducible representations

dim111111224
type+++++++
imageC1C2C2C2C4C4D4D4C4.D4
kernelC25.C4C24.4C4C2×C4.D4D4×C23C22×D4C25C22×C4C2×D4C22
# reps124144484

Matrix representation of C25.C4 ►in GL6(𝔽17)

1600000
0160000
000100
001000
0000016
0000160
,
100000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
0130000
1300000
000010
000001
000100
0016000

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C25.C4 in GAP, Magma, Sage, TeX 

C_2^5.C_4
 
% in TeX
 
G:=Group("C2^5.C4");
 
// GroupNames label
 
G:=SmallGroup(128,515);
 
// by ID
 
G=gap.SmallGroup(128,515);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=1,f^4=e,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*c*e,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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