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G = C22.138C25  order 128 = 27

119th central stem extension by C22 of C25

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

Aliases: C23.79C24, C22.138C25, C24.152C23, C42.121C23, C4.942+ 1+4, D42⋊24C2, (C4×D4)⋊69C22, C4⋊Q8⋊100C22, C4⋊C4.505C23, C4⋊1D4⋊28C22, C4⋊D4⋊93C22, (C2×C4).128C24, (C2×C42)⋊74C22, C22≀C2⋊19C22, C24⋊C22⋊7C2, (C2×D4).330C23, C4.4D4⋊42C22, (C22×D4)⋊48C22, C22⋊C4.53C23, (C2×Q8).308C23, C22.29C24⋊33C2, C42⋊C2⋊63C22, (C22×C4).398C23, C2.67(C2×2+ 1+4), C22.26C24⋊51C2, C22.49C24⋊24C2, (C2×C4○D4)⋊51C22, SmallGroup(128,2281)

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

Derived series
C1 — C22 — C22.138C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C22.26C24 — C22.138C25
Lower central
C1 — C22 — C22.138C25
Upper central
C1 — C22 — C22.138C25
Jennings
C1 — C22 — C22.138C25

Generators and relations for C22.138C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=f2=g2=a, e2=b, ab=ba, dcd-1=gcg-1=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: 1212 in 629 conjugacy classes, 384 normal (6 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C42, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×D4, C2×C4○D4, C22.26C24, C22.29C24, D42, C22.49C24, C24⋊C22, C22.138C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C22.138C25

Smallest permutation representation of C22.138C25
►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 24)(2 21)(3 22)(4 23)(5 10)(6 11)(7 12)(8 9)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2P4A···4F4G···4U
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim1111114
type+++++++
imageC1C2C2C2C2C22+ 1+4
kernelC22.138C25C22.26C24C22.29C24D42C22.49C24C24⋊C22C4
# reps13126646

Matrix representation of C22.138C25 ►in GL8(ℤ)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
0-1000000
-10000000
000-10000
00-100000
00000001
00000010
00000100
00001000
,
000-10000
00100000
0-1000000
10000000
0000000-1
00000010
00000-100
00001000
,
-10000000
0-1000000
00-100000
000-10000
00000010
00000001
0000-1000
00000-100
,
00100000
00010000
-10000000
0-1000000
00000010
00000001
0000-1000
00000-100
,
01000000
-10000000
00010000
00-100000
00000100
0000-1000
00000001
000000-10

G:=sub<GL(8,Integers())| [-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,-1,0,0,0,0,0,0,0,0,-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,-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,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0],[-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,-1,0,0,0,0,0,0,0,0,0,0,-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,-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],[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] >;
 

C22.138C25 in GAP, Magma, Sage, TeX 

C_2^2._{138}C_2^5
 
% in TeX
 
G:=Group("C2^2.138C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2281);
 
// by ID
 
G=gap.SmallGroup(128,2281);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,352,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=f^2=g^2=a,e^2=b,a*b=b*a,d*c*d^-1=g*c*g^-1=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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