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G = C22.123C25  order 128 = 27

104th central stem extension by C22 of C25

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

Aliases: C23.64C24, C22.123C25, C24.143C23, C42.106C23, C22.122+ 1+4, D42⋊19C2, D4⋊5D4⋊29C2, (C4×D4)⋊58C22, C4⋊1D4⋊22C22, C4⋊D4⋊33C22, C4⋊C4.311C23, C23⋊3D4⋊11C2, (C2×C4).113C24, (C23×C4)⋊49C22, C22⋊Q8⋊42C22, C22≀C2⋊40C22, (C2×D4).315C23, C4.4D4⋊34C22, (C22×D4)⋊42C22, C22⋊C4.41C23, (C2×Q8).300C23, C42.C2⋊17C22, C22.32C24⋊12C2, C22.19C24⋊37C2, C22.54C24⋊4C2, C42⋊2C2⋊10C22, C42⋊C2⋊51C22, (C22×C4).383C23, C22.45C24⋊14C2, C2.52(C2×2+ 1+4), C2.44(C2.C25), C22.56C24⋊3C2, C22.D4⋊60C22, C22.34C24⋊17C2, C22.31C24⋊19C2, C22.47C24⋊27C2, C22.33C24⋊11C2, (C2×C4⋊D4)⋊72C2, (C2×C4⋊C4)⋊83C22, (C2×C4○D4)⋊42C22, (C2×C22⋊C4)⋊58C22, (C2×C22.D4)⋊65C2, SmallGroup(128,2266)

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

Derived series
C1 — C22 — C22.123C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4⋊D4 — C22.123C25
Lower central
C1 — C22 — C22.123C25
Upper central
C1 — C22 — C22.123C25
Jennings
C1 — C22 — C22.123C25

Generators and relations for C22.123C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=g2=1, f2=a, ab=ba, dcd=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: 1044 in 583 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, C4○D4, C24, C24, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C23×C4, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C4⋊D4, C2×C22.D4, C22.19C24, C23⋊3D4, C23⋊3D4, C22.31C24, C22.32C24, C22.32C24, C22.33C24, C22.33C24, C22.34C24, D42, D4⋊5D4, C22.45C24, C22.47C24, C22.54C24, C22.56C24, C22.123C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.123C25

Smallest permutation representation of C22.123C25
►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 19)(6 20)(7 17)(8 18)(9 25)(10 26)(11 27)(12 28)(21 29)(22 30)(23 31)(24 32)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2P4A4B4C···4U
order122222222···2444···4
size111122224···4224···4

38 irreducible representations

dim11111111111111144
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.123C25C2×C4⋊D4C2×C22.D4C22.19C24C23⋊3D4C22.31C24C22.32C24C22.33C24C22.34C24D42D4⋊5D4C22.45C24C22.47C24C22.54C24C22.56C24C22C2
# reps11113133226222242

Matrix representation of C22.123C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
00040000
00100000
01000000
40000000
00000200
00003000
00000003
00000020
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
40000000
04000000
00400000
00040000
00000030
00000003
00002000
00000200
,
00100000
00010000
40000000
04000000
00000010
00000001
00004000
00000400
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010

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

C22.123C25 in GAP, Magma, Sage, TeX 

C_2^2._{123}C_2^5
 
% in TeX
 
G:=Group("C2^2.123C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2266);
 
// by ID
 
G=gap.SmallGroup(128,2266);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,2019,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=g^2=1,f^2=a,a*b=b*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=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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