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G = C22.134C25  order 128 = 27

115th central stem extension by C22 of C25

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

Aliases: C23.75C24, C22.134C25, C24.150C23, C42.117C23, C22.172+ 1+4, C4⋊Q8⋊45C22, D4⋊5D4⋊35C2, (C4×D4)⋊67C22, (C4×Q8)⋊64C22, C4⋊C4.321C23, C4⋊D4⋊40C22, C23⋊3D4⋊15C2, (C2×C4).124C24, (C2×C42)⋊72C22, C22⋊Q8⋊50C22, C22≀C2⋊17C22, C24⋊C22⋊6C2, (C2×D4).326C23, C4.4D4⋊41C22, (C22×D4)⋊47C22, C22⋊C4.49C23, (C2×Q8).306C23, C42.C2⋊66C22, (C22×Q8)⋊43C22, C42⋊2C2⋊44C22, C22.32C24⋊19C2, C42⋊C2⋊61C22, (C22×C4).394C23, C22.45C24⋊19C2, C2.63(C2×2+ 1+4), C2.50(C2.C25), C22.56C24⋊8C2, C22.57C24⋊9C2, C22.D4⋊21C22, C23.36C23⋊48C2, C22.49C24⋊22C2, C23.38C23⋊31C2, C22.36C24⋊31C2, (C2×C4.4D4)⋊59C2, (C2×C22⋊C4)⋊63C22, (C2×C4○D4).240C22, SmallGroup(128,2277)

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

Derived series
C1 — C22 — C22.134C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4.4D4 — C22.134C25
Lower central
C1 — C22 — C22.134C25
Upper central
C1 — C22 — C22.134C25
Jennings
C1 — C22 — C22.134C25

Generators and relations for C22.134C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=ba=ab, f2=a, dcd=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 876 in 530 conjugacy classes, 380 normal (24 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4.4D4, C23.36C23, C23⋊3D4, C23.38C23, C22.32C24, C22.36C24, D4⋊5D4, C22.45C24, C22.49C24, C24⋊C22, C22.56C24, C22.57C24, C22.134C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.134C25

Smallest permutation representation of C22.134C25
►On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)

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

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A4B4C4D4E···4X
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim111111111111144
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.134C25C2×C4.4D4C23.36C23C23⋊3D4C23.38C23C22.32C24C22.36C24D4⋊5D4C22.45C24C22.49C24C24⋊C22C22.56C24C22.57C24C22C2
# reps112224446212124

Matrix representation of C22.134C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
24000000
33000000
02030000
33200000
00000040
00001413
00004000
00004041
,
40030000
00110000
01040000
00010000
00000400
00004000
00001413
00001404
,
30000000
03000000
00300000
00030000
00000010
00001413
00004000
00004101
,
10300000
00110000
10400000
44100000
00000100
00001000
00001413
00000004
,
43000000
01000000
04010000
01100000
00004000
00000400
00000040
00000004

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,3,0,3,0,0,0,0,4,3,2,3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,4,0,0,0,0,0,0,4,1,0,4,0,0,0,0,0,3,0,1],[4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,4,1,0,0,0,0,0,0,0,0,0,4,1,1,0,0,0,0,4,0,4,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,4,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,1],[1,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,3,1,4,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4],[4,0,0,0,0,0,0,0,3,1,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
 

C22.134C25 in GAP, Magma, Sage, TeX 

C_2^2._{134}C_2^5
 
% in TeX
 
G:=Group("C2^2.134C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2277);
 
// by ID
 
G=gap.SmallGroup(128,2277);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,520,2019,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=b*a=a*b,f^2=a,d*c*d=g*c*g=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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