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G = C22.99C25  order 128 = 27

80th central stem extension by C22 of C25

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

Aliases: C42.91C23, C22.99C25, C24.139C23, C23.140C24, C4⋊Q8⋊34C22, (C4×D4)⋊49C22, (C2×C4).89C24, (C4×Q8)⋊48C22, C4⋊C4.494C23, (C2×C42)⋊63C22, C22⋊Q8⋊35C22, (C2×D4).306C23, C4.4D4⋊28C22, (C2×Q8).291C23, C42.C2⋊75C22, C22.45C24⋊7C2, C42⋊C2⋊42C22, C22.11C24⋊20C2, C42⋊2C2⋊37C22, C22≀C2.10C22, C4⋊1D4.113C22, C4⋊D4.226C22, C22⋊C4.108C23, (C22×C4).369C23, C2.31(C2.C25), C22.29C24.16C2, (C22×D4).428C22, C22.D4⋊53C22, (C22×Q8).362C22, C22.50C24⋊23C2, C23.36C23⋊31C2, C22.53C24⋊14C2, C23.38C23⋊25C2, C23.32C23⋊15C2, C23.37C23⋊42C2, C22.36C24⋊14C2, (C4×C4○D4)⋊30C2, C4.182(C2×C4○D4), (C2×C4.4D4)⋊55C2, C2.55(C22×C4○D4), C22.32(C2×C4○D4), (C2×C4).308(C4○D4), (C2×C4○D4).331C22, (C2×C22⋊C4).384C22, SmallGroup(128,2242)

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

Derived series
C1 — C22 — C22.99C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.99C25
Lower central
C1 — C22 — C22.99C25
Upper central
C1 — C22 — C22.99C25
Jennings
C1 — C22 — C22.99C25

Subgroups: 748 in 520 conjugacy classes, 388 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×22], C22, C22 [×2], C22 [×24], C2×C4 [×2], C2×C4 [×26], C2×C4 [×22], D4 [×22], Q8 [×14], C23, C23 [×6], C23 [×8], C42 [×2], C42 [×20], C22⋊C4 [×50], C4⋊C4 [×38], C22×C4, C22×C4 [×18], C2×D4, C2×D4 [×14], C2×D4 [×4], C2×Q8, C2×Q8 [×8], C2×Q8 [×4], C4○D4 [×4], C24 [×2], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×8], C42⋊C2 [×3], C42⋊C2 [×14], C4×D4 [×22], C4×Q8 [×14], C22≀C2 [×4], C4⋊D4 [×4], C22⋊Q8 [×16], C22.D4 [×20], C4.4D4 [×20], C42.C2 [×2], C42⋊2C2 [×16], C4⋊1D4 [×2], C4⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C4×C4○D4, C22.11C24, C23.32C23, C2×C4.4D4, C23.36C23 [×4], C23.37C23, C22.29C24, C23.38C23, C22.36C24 [×4], C22.45C24 [×8], C22.50C24 [×4], C22.53C24 [×4], C22.99C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], C25, C22×C4○D4, C2.C25 [×2], C22.99C25

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

Smallest permutation representation
►On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

(1 9)(2 16)(3 11)(4 14)(5 18)(6 21)(7 20)(8 23)(10 28)(12 26)(13 25)(15 27)(17 29)(19 31)(22 32)(24 30)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)

(1 29 25 6)(2 30 26 7)(3 31 27 8)(4 32 28 5)(9 19 13 23)(10 20 14 24)(11 17 15 21)(12 18 16 22)

(2 28)(4 26)(5 30)(7 32)(10 16)(12 14)(18 24)(20 22)

(1 15 27 9)(2 16 28 10)(3 13 25 11)(4 14 26 12)(5 20 30 22)(6 17 31 23)(7 18 32 24)(8 19 29 21)
 
G:=sub<Sym(32)| (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,9)(2,16)(3,11)(4,14)(5,18)(6,21)(7,20)(8,23)(10,28)(12,26)(13,25)(15,27)(17,29)(19,31)(22,32)(24,30), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,29,25,6)(2,30,26,7)(3,31,27,8)(4,32,28,5)(9,19,13,23)(10,20,14,24)(11,17,15,21)(12,18,16,22), (2,28)(4,26)(5,30)(7,32)(10,16)(12,14)(18,24)(20,22), (1,15,27,9)(2,16,28,10)(3,13,25,11)(4,14,26,12)(5,20,30,22)(6,17,31,23)(7,18,32,24)(8,19,29,21)>;
 
G:=Group( (1,27)(2,28)(3,25)(4,26)(5,30)(6,31)(7,32)(8,29)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32), (1,9)(2,16)(3,11)(4,14)(5,18)(6,21)(7,20)(8,23)(10,28)(12,26)(13,25)(15,27)(17,29)(19,31)(22,32)(24,30), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,29,25,6)(2,30,26,7)(3,31,27,8)(4,32,28,5)(9,19,13,23)(10,20,14,24)(11,17,15,21)(12,18,16,22), (2,28)(4,26)(5,30)(7,32)(10,16)(12,14)(18,24)(20,22), (1,15,27,9)(2,16,28,10)(3,13,25,11)(4,14,26,12)(5,20,30,22)(6,17,31,23)(7,18,32,24)(8,19,29,21) );
 
G=PermutationGroup([(1,27),(2,28),(3,25),(4,26),(5,30),(6,31),(7,32),(8,29),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32)], [(1,9),(2,16),(3,11),(4,14),(5,18),(6,21),(7,20),(8,23),(10,28),(12,26),(13,25),(15,27),(17,29),(19,31),(22,32),(24,30)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,29,25,6),(2,30,26,7),(3,31,27,8),(4,32,28,5),(9,19,13,23),(10,20,14,24),(11,17,15,21),(12,18,16,22)], [(2,28),(4,26),(5,30),(7,32),(10,16),(12,14),(18,24),(20,22)], [(1,15,27,9),(2,16,28,10),(3,13,25,11),(4,14,26,12),(5,20,30,22),(6,17,31,23),(7,18,32,24),(8,19,29,21)])
 

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
240000
000100
001000
000404
004040
,
300000
030000
002040
000204
003030
000303
,
320000
020000
002000
000200
000020
000002
,
400000
040000
001000
000100
004040
000404
,
400000
040000
000100
004000
000001
000040

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],[1,2,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,4,0,0,1,0,4,0,0,0,0,0,0,4,0,0,0,0,4,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,3,0,0,0,0,2,0,3,0,0,4,0,3,0,0,0,0,4,0,3],[3,0,0,0,0,0,2,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,4,0,0,0,0,1,0,4,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
 

44 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A···4P4Q···4AF
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim111111111111124
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.99C25C4×C4○D4C22.11C24C23.32C23C2×C4.4D4C23.36C23C23.37C23C22.29C24C23.38C23C22.36C24C22.45C24C22.50C24C22.53C24C2×C4C2
# reps111114111484484

In GAP, Magma, Sage, TeX 

C_2^2._{99}C_2^5
 
% in TeX
 
G:=Group("C2^2.99C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2242);
 
// by ID
 
G=gap.SmallGroup(128,2242);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,520,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=b,e^2=b*a=a*b,g^2=a,d*c*d^-1=g*c*g^-1=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^-1=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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