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G = C22.79C25  order 128 = 27

60th central stem extension by C22 of C25

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

Aliases: C42⋊3C23, C25.78C22, C22.79C25, C23.131C24, C24.132C23, C22.72+ 1+4, C4⋊C4⋊21C23, (C2×D4)⋊8C23, D4⋊5D4⋊16C2, (C2×Q8)⋊8C23, (C4×D4)⋊37C22, C23⋊6(C4○D4), C23⋊3D4⋊6C2, (C2×C4).72C24, (C22×C4)⋊4C23, C23⋊2Q8⋊4C2, C22≀C2⋊6C22, C4⋊D4⋊22C22, C22⋊C4⋊20C23, (C23×C4)⋊41C22, C22⋊Q8⋊25C22, C22.32C24⋊2C2, C42⋊2C2⋊1C22, C4.4D4⋊25C22, (C22×D4)⋊35C22, C22.11C24⋊15C2, C42⋊C2⋊34C22, C22.45C24⋊2C2, C22.19C24⋊26C2, C2.28(C2×2+ 1+4), C22.D4⋊50C22, C22⋊C4○C22≀C2, (C2×C22≀C2)⋊26C2, (C2×C4○D4)⋊27C22, C2.44(C22×C4○D4), C22.26(C2×C4○D4), (C22×C22⋊C4)⋊34C2, (C2×C22⋊C4)⋊46C22, SmallGroup(128,2222)

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

Derived series
C1 — C22 — C22.79C25
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C22.79C25
Lower central
C1 — C22 — C22.79C25
Upper central
C1 — C22 — C22.79C25
Jennings
C1 — C22 — C22.79C25

Generators and relations for C22.79C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=b, ab=ba, dcd-1=gcg=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1196 in 664 conjugacy classes, 392 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, C24, C2×C22⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C23×C4, C22×D4, C2×C4○D4, C25, C22×C22⋊C4, C22.11C24, C2×C22≀C2, C22.19C24, C23⋊3D4, C22.32C24, C23⋊2Q8, D4⋊5D4, C22.45C24, C22.79C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C25, C22×C4○D4, C2×2+ 1+4, C22.79C25

Permutation representations of C22.79C25
►On 16 points - transitive group 16T201
Generators in S16
(1 9)(2 10)(3 11)(4 12)(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 5)(2 15)(3 7)(4 13)(6 10)(8 12)(9 14)(11 16)

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

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

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

(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)
 
G:=sub<Sym(16)| (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,5)(2,15)(3,7)(4,13)(6,10)(8,12)(9,14)(11,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,12)(4,10)(5,7)(6,13)(8,15)(9,11)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12)>;
 
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,5)(2,15)(3,7)(4,13)(6,10)(8,12)(9,14)(11,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11)(2,12)(3,9)(4,10)(5,14)(6,15)(7,16)(8,13), (1,3)(2,12)(4,10)(5,7)(6,13)(8,15)(9,11)(14,16), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12) );
 
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,5),(2,15),(3,7),(4,13),(6,10),(8,12),(9,14),(11,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,11),(2,12),(3,9),(4,10),(5,14),(6,15),(7,16),(8,13)], [(1,3),(2,12),(4,10),(5,7),(6,13),(8,15),(9,11),(14,16)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,11),(10,12)]])
 
G:=TransitiveGroup(16,201);
 

44 conjugacy classes

class 1 2A2B2C2D···2M2N···2S4A···4H4I···4X
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC22.79C25C22×C22⋊C4C22.11C24C2×C22≀C2C22.19C24C23⋊3D4C22.32C24C23⋊2Q8D4⋊5D4C22.45C24C23C22
# reps112241418884

Matrix representation of C22.79C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
130000
040000
000010
000001
001000
000100
,
300000
030000
000100
001000
000004
000040
,
400000
410000
001000
000100
000010
000001
,
400000
040000
001000
000400
000010
000004
,
100000
010000
001000
000100
000040
000004

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

C22.79C25 in GAP, Magma, Sage, TeX 

C_2^2._{79}C_2^5
 
% in TeX
 
G:=Group("C2^2.79C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2222);
 
// by ID
 
G=gap.SmallGroup(128,2222);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,570,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=b,a*b=b*a,d*c*d^-1=g*c*g=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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