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G = C22.124C25  order 128 = 27

105th central stem extension by C22 of C25

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

Aliases: C23.65C24, C22.124C25, C42.107C23, C24.144C23, C22.92- 1+4, C22.132+ 1+4, C4⋊Q8⋊38C22, D4⋊6D4⋊32C2, D4⋊5D4⋊30C2, D4⋊3Q8⋊32C2, (C4×D4)⋊59C22, (C4×Q8)⋊56C22, C23⋊2Q8⋊7C2, C4⋊C4.312C23, C4⋊D4⋊34C22, (C2×C4).114C24, C22⋊Q8⋊43C22, (C2×D4).316C23, C4.4D4⋊35C22, C22⋊C4.42C23, (C2×Q8).301C23, C42.C2⋊18C22, (C22×Q8)⋊38C22, C42⋊2C2⋊11C22, C42⋊C2⋊52C22, C22.19C24⋊38C2, C22.32C24⋊13C2, C22≀C2.12C22, (C22×C4).384C23, (C23×C4).616C22, C22.45C24⋊15C2, C2.53(C2×2+ 1+4), C2.39(C2×2- 1+4), C2.45(C2.C25), C22.56C24⋊4C2, C22.57C24⋊5C2, (C22×D4).436C22, C22.D4⋊14C22, C23.38C23⋊26C2, C22.33C24⋊12C2, C22.36C24⋊25C2, C23.41C23⋊18C2, C22.46C24⋊28C2, (C2×C4⋊C4)⋊84C22, (C2×C22⋊Q8)⋊82C2, (C2×C4○D4)⋊43C22, (C2×C22.D4)⋊66C2, (C2×C22⋊C4).389C22, SmallGroup(128,2267)

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

Derived series
C1 — C22 — C22.124C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C22⋊Q8 — C22.124C25
Lower central
C1 — C22 — C22.124C25
Upper central
C1 — C22 — C22.124C25
Jennings
C1 — C22 — C22.124C25

Generators and relations for C22.124C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=g2=1, d2=f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece=fcf-1=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 788 in 519 conjugacy classes, 382 normal (58 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C22⋊Q8, C2×C22.D4, C22.19C24, C23.38C23, C22.32C24, C22.33C24, C22.33C24, C22.36C24, C23⋊2Q8, C23.41C23, D4⋊5D4, D4⋊6D4, C22.45C24, C22.46C24, D4⋊3Q8, C22.56C24, C22.57C24, C22.124C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C2.C25, C22.124C25

Smallest permutation representation of C22.124C25
►On 32 points
Generators in S32
(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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)

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

(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)

(2 28)(4 26)(6 17)(8 19)(10 30)(12 32)(14 22)(16 24)

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2L4A4B4C···4Y
order122222222···2444···4
size111122224···4224···4

38 irreducible representations

dim11111111111111111444
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+42- 1+4C2.C25
kernelC22.124C25C2×C22⋊Q8C2×C22.D4C22.19C24C23.38C23C22.32C24C22.33C24C22.36C24C23⋊2Q8C23.41C23D4⋊5D4D4⋊6D4C22.45C24C22.46C24D4⋊3Q8C22.56C24C22.57C24C22C22C2
# reps11112152112242222222

Matrix representation of C22.124C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
00100000
00040000
10000000
04000000
00000010
00000442
00001000
00003021
,
00100000
00010000
40000000
04000000
00000010
00000103
00001000
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000040
00000104
,
20000000
02000000
00300000
00030000
00002400
00003300
00000422
00004013
,
01000000
10000000
00010000
00100000
00001000
00000100
00000010
00000001

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

C22.124C25 in GAP, Magma, Sage, TeX 

C_2^2._{124}C_2^5
 
% in TeX
 
G:=Group("C2^2.124C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2267);
 
// by ID
 
G=gap.SmallGroup(128,2267);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,2019,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=g^2=1,d^2=f^2=a,a*b=b*a,d*c*d^-1=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=f*c*f^-1=b*c=c*b,e*d*e=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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