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G = C22.131C25  order 128 = 27

112nd central stem extension by C22 of C25

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

Aliases: C23.72C24, C24.148C23, C22.131C25, C42.114C23, C4.902+ (1+4), C22.142+ (1+4), (D42)⋊21C2, C4⋊Q8⋊44C22, D4⋊5D4⋊33C2, D4⋊6D4⋊36C2, Q8⋊6D4⋊26C2, (C4×D4)⋊65C22, (C4×Q8)⋊62C22, C4⋊D4⋊92C22, C23⋊3D4⋊13C2, C4⋊C4.319C23, C4⋊1D4⋊25C22, (C2×C4).121C24, (C23×C4)⋊52C22, C22⋊Q8⋊49C22, C22≀C2⋊15C22, (C2×D4).323C23, C4.4D4⋊40C22, (C22×D4)⋊45C22, (C2×Q8).464C23, C42.C2⋊24C22, C22.54C24⋊8C2, C42⋊2C2⋊18C22, C22.29C24⋊30C2, C42⋊C2⋊59C22, C22.19C24⋊44C2, C22.32C24⋊18C2, C22⋊C4.115C23, (C22×C4).391C23, C22.45C24⋊18C2, C2.60(C2×2+ (1+4)), C2.49(C2.C25), C22.56C24⋊7C2, C22.D4⋊20C22, C22.36C24⋊29C2, C22.46C24⋊32C2, C22.31C24⋊23C2, C22.47C24⋊31C2, C22.34C24⋊20C2, C22.49C24⋊20C2, (C2×C4⋊D4)⋊75C2, (C2×C4⋊C4)⋊89C22, (C2×C4○D4)⋊48C22, (C2×C22⋊C4)⋊61C22, SmallGroup(128,2274)

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

Derived series
C1 — C22 — C22.131C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.131C25
Lower central
C1 — C22 — C22.131C25
Upper central
C1 — C22 — C22.131C25
Jennings
C1 — C22 — C22.131C25

Subgroups: 1028 in 580 conjugacy classes, 382 normal (122 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×19], C22, C22 [×2], C22 [×44], C2×C4 [×10], C2×C4 [×10], C2×C4 [×29], D4 [×47], Q8 [×5], C23 [×5], C23 [×6], C23 [×18], C42 [×2], C42 [×6], C22⋊C4 [×10], C22⋊C4 [×38], C4⋊C4 [×10], C4⋊C4 [×18], C22×C4 [×8], C22×C4 [×14], C22×C4 [×2], C2×D4 [×10], C2×D4 [×30], C2×D4 [×12], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×10], C24, C24 [×4], C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C42⋊C2 [×3], C42⋊C2 [×4], C4×D4 [×7], C4×D4 [×8], C4×Q8, C22≀C2 [×2], C22≀C2 [×14], C4⋊D4 [×11], C4⋊D4 [×28], C22⋊Q8 [×5], C22⋊Q8 [×6], C22.D4 [×2], C22.D4 [×20], C4.4D4, C4.4D4 [×8], C42.C2, C42.C2 [×2], C42⋊2C2 [×6], C4⋊1D4, C4⋊1D4 [×4], C4⋊Q8, C23×C4, C22×D4 [×2], C22×D4 [×4], C2×C4○D4 [×3], C2×C4○D4 [×2], C2×C4⋊D4, C22.19C24 [×2], C23⋊3D4 [×2], C22.29C24, C22.29C24 [×2], C22.31C24, C22.32C24 [×2], C22.34C24, C22.34C24 [×2], C22.36C24, D42, D4⋊5D4 [×2], D4⋊5D4 [×2], D4⋊6D4, Q8⋊6D4, C22.45C24 [×2], C22.46C24, C22.47C24, C22.49C24, C22.54C24 [×2], C22.56C24 [×2], C22.131C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×4], C25, C2×2+ (1+4) [×2], C2.C25, C22.131C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=d2=e2=1, c2=f2=g2=a, ab=ba, dcd=gcg-1=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 >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
000-10000
00100000
01000000
-10000000
000010-12
000000-11
0000-221-2
0000-211-2
,
01000000
10000000
000-10000
00-100000
0000-10-10
000000-11
00000010
00000110
,
00100000
00010000
10000000
01000000
00001000
00000100
00000010
00000001
,
01000000
10000000
00010000
00100000
0000-10-10
000000-11
00002010
00002-110
,
-10000000
0-1000000
00-100000
000-10000
00001-200
00001-100
000002-12
0000-12-11

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],[0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,-2,-2,0,0,0,0,0,0,2,1,0,0,0,0,-1,-1,1,1,0,0,0,0,2,1,-2,-2],[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,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,1,1,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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,-1,0,2,2,0,0,0,0,0,0,0,-1,0,0,0,0,-1,-1,1,1,0,0,0,0,0,1,0,0],[-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,-1,0,0,0,0,-2,-1,2,2,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,2,1] >;
 

38 conjugacy classes

class 1 2A2B2C2D2E2F···2O4A4B4C4D4E···4V
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim1111111111111111111444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ (1+4)2+ (1+4)C2.C25
kernelC22.131C25C2×C4⋊D4C22.19C24C23⋊3D4C22.29C24C22.31C24C22.32C24C22.34C24C22.36C24D42D4⋊5D4D4⋊6D4Q8⋊6D4C22.45C24C22.46C24C22.47C24C22.49C24C22.54C24C22.56C24C4C22C2
# reps1122312311411211122222

In GAP, Magma, Sage, TeX 

C_2^2._{131}C_2^5
 
% in TeX
 
G:=Group("C2^2.131C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2274);
 
// by ID
 
G=gap.SmallGroup(128,2274);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,723,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=e^2=1,c^2=f^2=g^2=a,a*b=b*a,d*c*d=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=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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