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G = C22.122C25  order 128 = 27

103rd central stem extension by C22 of C25

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

Aliases: C23.63C24, C22.122C25, C42.105C23, C24.142C23, C22.112+ 1+4, C4⋊Q8⋊37C22, D4⋊5D4⋊28C2, D4⋊6D4⋊31C2, (C4×D4)⋊57C22, (C4×Q8)⋊55C22, C4⋊1D4⋊21C22, C4⋊D4⋊90C22, C23⋊3D4⋊10C2, C4⋊C4.310C23, (C2×C4).112C24, (C23×C4)⋊48C22, C22⋊Q8⋊41C22, C22≀C2⋊12C22, C42⋊2C2⋊9C22, (C2×D4).314C23, C4.4D4⋊33C22, C22⋊C4.40C23, (C2×Q8).299C23, C42.C2⋊16C22, C22.19C24⋊36C2, C22.32C24⋊11C2, C22.29C24⋊27C2, C42⋊C2⋊50C22, C22.54C24⋊3C2, (C22×C4).382C23, C22.45C24⋊13C2, C2.51(C2×2+ 1+4), C2.43(C2.C25), C22.56C24⋊2C2, C22.57C24⋊4C2, (C22×D4).435C22, C22.D4⋊13C22, C22.33C24⋊10C2, C22.36C24⋊24C2, C22.46C24⋊27C2, C22.34C24⋊16C2, C22.53C24⋊19C2, C22.47C24⋊26C2, (C2×C4⋊C4)⋊82C22, (C2×C4○D4)⋊41C22, (C2×C22⋊C4)⋊57C22, (C2×C22.D4)⋊64C2, SmallGroup(128,2265)

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

Derived series
C1 — C22 — C22.122C25
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C22.D4 — C22.122C25
Lower central
C1 — C22 — C22.122C25
Upper central
C1 — C22 — C22.122C25
Jennings
C1 — C22 — C22.122C25

Generators and relations for C22.122C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=a, ab=ba, dcd=gcg=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 860 in 530 conjugacy classes, 380 normal (56 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C23×C4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C22.D4, C22.19C24, C23⋊3D4, C22.29C24, C22.32C24, C22.33C24, C22.34C24, C22.36C24, D4⋊5D4, D4⋊6D4, C22.45C24, C22.45C24, C22.46C24, C22.47C24, C22.53C24, C22.54C24, C22.56C24, C22.57C24, C22.122C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.122C25

Smallest permutation representation of C22.122C25
►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 15)(2 16)(3 13)(4 14)(5 20)(6 17)(7 18)(8 19)(9 24)(10 21)(11 22)(12 23)(25 30)(26 31)(27 32)(28 29)

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

(2 16)(4 14)(5 7)(6 19)(8 17)(9 24)(11 22)(18 20)(25 32)(26 28)(27 30)(29 31)

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111111111144
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.122C25C2×C22.D4C22.19C24C23⋊3D4C22.29C24C22.32C24C22.33C24C22.34C24C22.36C24D4⋊5D4D4⋊6D4C22.45C24C22.46C24C22.47C24C22.53C24C22.54C24C22.56C24C22.57C24C22C2
# reps11211422222322112124

Matrix representation of C22.122C25 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01300000
10030000
00040000
00400000
00000010
00000103
00001000
00000004
,
10000000
04000000
04100000
10040000
00001000
00003400
00000040
00004441
,
01000000
10000000
10040000
01400000
00002200
00000300
00000334
00000302
,
10000000
01000000
01400000
10040000
00001100
00000400
00000442
00000401
,
10000000
01000000
00100000
00010000
00001000
00000100
00000040
00000104

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

C22.122C25 in GAP, Magma, Sage, TeX 

C_2^2._{122}C_2^5
 
% in TeX
 
G:=Group("C2^2.122C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2265);
 
// by ID
 
G=gap.SmallGroup(128,2265);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,184,2019,570,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=g^2=1,e^2=a,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^-1=f*c*f=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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