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G = C22.138C25order 128 = 27

119th central stem extension by C22 of C25

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

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 [×3], C2 [×13], C4 [×6], C4 [×15], C22, C22 [×55], C2×C4 [×18], C2×C4 [×30], D4 [×66], Q8 [×6], C23, C23 [×12], C23 [×24], C42, C42 [×11], C22⋊C4 [×48], C4⋊C4 [×12], C22×C4 [×15], C2×D4 [×54], C2×D4 [×24], C2×Q8 [×6], C4○D4 [×12], C24 [×8], C2×C42, C42⋊C2 [×12], C4×D4 [×12], C22≀C2 [×32], C4⋊D4 [×36], C4.4D4 [×30], C41D4 [×11], C4⋊Q8 [×3], C22×D4 [×12], C2×C4○D4 [×6], C22.26C24 [×3], C22.29C24 [×12], D42 [×6], C22.49C24 [×6], C24⋊C22 [×4], C22.138C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ 1+4 [×6], C25, C2×2+ 1+4 [×3], 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 2A 2B 2C 2D ··· 2P 4A ··· 4F 4G ··· 4U order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 1 1 4 ··· 4 2 ··· 2 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 2+ 1+4 kernel C22.138C25 C22.26C24 C22.29C24 D42 C22.49C24 C24⋊C22 C4 # reps 1 3 12 6 6 4 6

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

 -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

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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