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G = C22.73C25  order 128 = 27

54th central stem extension by C22 of C25

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

Aliases: C24⋊3C23, C42⋊2C23, C25⋊6C22, C23.33C24, C22.73C25, C22⋊12+ (1+4), D4○C22≀C2, (D42)⋊11C2, (C2×D4)⋊53D4, D4⋊13(C2×D4), C4⋊C4⋊8C23, C23⋊7(C2×D4), D4⋊5D4⋊12C2, (C2×D4)⋊19C23, (C4×D4)⋊32C22, (D4×C23)⋊16C2, C23⋊3D4⋊4C2, C22⋊C4⋊9C23, (C2×C4).67C24, (C22×C4)⋊3C23, (C2×Q8)⋊18C23, C2.25(D4×C23), C22≀C2⋊3C22, C4⋊1D4⋊15C22, C4⋊D4⋊78C22, (C23×C4)⋊38C22, C4.114(C22×D4), C22⋊Q8⋊93C22, C22.9(C22×D4), C4.4D4⋊19C22, (C22×D4)⋊32C22, (C2×2+ (1+4))⋊7C2, C22.19C24⋊20C2, C22.11C24⋊13C2, C22.29C24⋊18C2, C42⋊C2⋊30C22, C2.26(C2×2+ (1+4)), C22.D4⋊48C22, (C2×C4)⋊12(C2×D4), (C2×D4)○C22≀C2, (C2×C22≀C2)⋊25C2, (C2×C4○D4)⋊22C22, (C2×C22⋊C4)⋊43C22, SmallGroup(128,2216)

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

Derived series
C1 — C22 — C22.73C25
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C22.73C25
Lower central
C1 — C22 — C22.73C25
Upper central
C1 — C22 — C22.73C25
Jennings
C1 — C22 — C22.73C25

Subgroups: 1836 in 950 conjugacy classes, 432 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×22], C4 [×4], C4 [×14], C22, C22 [×14], C22 [×102], C2×C4 [×20], C2×C4 [×34], D4 [×16], D4 [×84], Q8 [×4], C23, C23 [×22], C23 [×106], C42 [×4], C22⋊C4 [×44], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×20], C22×C4 [×4], C2×D4 [×62], C2×D4 [×92], C2×Q8 [×2], C4○D4 [×24], C24, C24 [×16], C24 [×16], C2×C22⋊C4 [×12], C42⋊C2 [×2], C4×D4 [×16], C22≀C2 [×36], C4⋊D4 [×28], C22⋊Q8 [×4], C22.D4 [×12], C4.4D4 [×4], C4⋊1D4 [×4], C23×C4, C22×D4 [×2], C22×D4 [×26], C22×D4 [×8], C2×C4○D4 [×6], 2+ (1+4) [×8], C25 [×2], C22.11C24, C2×C22≀C2 [×4], C22.19C24 [×2], C23⋊3D4 [×4], C22.29C24 [×2], D42 [×8], D4⋊5D4 [×8], D4×C23, C2×2+ (1+4), C22.73C25

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2+ (1+4) [×4], C25, D4×C23, C2×2+ (1+4) [×2], C22.73C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, dcd=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 >

Permutation representations
►On 16 points - transitive group 16T198
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)

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

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

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

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

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

(1 13)(2 14)(3 6)(4 5)(7 12)(8 11)(9 15)(10 16)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,13)(2,14)(3,6)(4,5)(7,11)(8,12)(9,16)(10,15), (1,9)(2,10)(3,8)(4,7)(5,11)(6,12)(13,16)(14,15), (1,3)(2,4)(5,14)(6,13)(7,9)(8,10)(11,16)(12,15), (1,14)(2,13)(3,5)(4,6)(7,8)(9,10)(11,12)(15,16), (1,13)(2,14)(3,5)(4,6)(7,12)(8,11)(9,16)(10,15), (1,13)(2,14)(3,6)(4,5)(7,12)(8,11)(9,15)(10,16)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,13)(2,14)(3,6)(4,5)(7,11)(8,12)(9,16)(10,15), (1,9)(2,10)(3,8)(4,7)(5,11)(6,12)(13,16)(14,15), (1,3)(2,4)(5,14)(6,13)(7,9)(8,10)(11,16)(12,15), (1,14)(2,13)(3,5)(4,6)(7,8)(9,10)(11,12)(15,16), (1,13)(2,14)(3,5)(4,6)(7,12)(8,11)(9,16)(10,15), (1,13)(2,14)(3,6)(4,5)(7,12)(8,11)(9,15)(10,16) );
 
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,13),(2,14),(3,6),(4,5),(7,11),(8,12),(9,16),(10,15)], [(1,9),(2,10),(3,8),(4,7),(5,11),(6,12),(13,16),(14,15)], [(1,3),(2,4),(5,14),(6,13),(7,9),(8,10),(11,16),(12,15)], [(1,14),(2,13),(3,5),(4,6),(7,8),(9,10),(11,12),(15,16)], [(1,13),(2,14),(3,5),(4,6),(7,12),(8,11),(9,16),(10,15)], [(1,13),(2,14),(3,6),(4,5),(7,12),(8,11),(9,15),(10,16)])
 
G:=TransitiveGroup(16,198);
 

Matrix representation ►G ⊆ GL6(ℤ)

100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
010000
100000
000010
000001
001000
000100
,
100000
010000
000100
001000
00000-1
0000-10
,
-100000
010000
001000
000100
000010
000001
,
-100000
0-10000
001000
000-100
000010
00000-1
,
100000
010000
001000
000100
0000-10
00000-1

G:=sub<GL(6,Integers())| [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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[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,-1,0,0,0,0,0,0,-1] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2Q2R···2Y4A4B4C4D4E···4R
order12222···22···244444···4
size11112···24···422224···4

44 irreducible representations

dim111111111124
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D42+ (1+4)
kernelC22.73C25C22.11C24C2×C22≀C2C22.19C24C23⋊3D4C22.29C24D42D4⋊5D4D4×C23C2×2+ (1+4)C2×D4C22
# reps114242881184

In GAP, Magma, Sage, TeX 

C_2^2._{73}C_2^5
 
% in TeX
 
G:=Group("C2^2.73C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2216);
 
// by ID
 
G=gap.SmallGroup(128,2216);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,a*b=b*a,d*c*d=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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