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G = C22.132C25  order 128 = 27

113rd central stem extension by C22 of C25

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

Aliases: C23.73C24, C22.132C25, C42.115C23, C24.149C23, C4.1602+ (1+4), C22.152+ (1+4), (D42)⋊22C2, D4⋊5D4⋊34C2, Q8⋊6D4⋊27C2, (C4×D4)⋊66C22, (C4×Q8)⋊63C22, C23⋊3D4⋊14C2, C4⋊1D4⋊26C22, C4⋊C4.320C23, C4⋊D4⋊39C22, (C2×C4).122C24, (C2×C42)⋊71C22, C22≀C2⋊16C22, (C2×D4).324C23, C4.4D4⋊91C22, (C22×D4)⋊46C22, C22⋊C4.47C23, C22⋊Q8⋊101C22, (C2×Q8).465C23, C42.C2⋊65C22, C42⋊2C2⋊43C22, C22.29C24⋊31C2, C22.54C24⋊9C2, C42⋊C2⋊60C22, (C22×C4).392C23, C2.61(C2×2+ (1+4)), C22.D4⋊62C22, C22.34C24⋊21C2, C23.36C23⋊46C2, C22.53C24⋊21C2, (C2×C4⋊1D4)⋊29C2, (C2×C4○D4)⋊49C22, (C2×C22⋊C4)⋊62C22, SmallGroup(128,2275)

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

Derived series
C1 — C22 — C22.132C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4⋊1D4 — C22.132C25
Lower central
C1 — C22 — C22.132C25
Upper central
C1 — C22 — C22.132C25
Jennings
C1 — C22 — C22.132C25

Subgroups: 1212 in 630 conjugacy classes, 384 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×4], C4 [×16], C22, C22 [×2], C22 [×54], C2×C4 [×2], C2×C4 [×16], C2×C4 [×26], D4 [×68], Q8 [×4], C23, C23 [×12], C23 [×28], C42 [×2], C42 [×6], C22⋊C4 [×44], C4⋊C4 [×20], C22×C4, C22×C4 [×18], C2×D4 [×54], C2×D4 [×28], C2×Q8 [×2], C4○D4 [×8], C24 [×8], C2×C42, C2×C22⋊C4 [×4], C42⋊C2 [×4], C4×D4 [×14], C4×Q8 [×2], C22≀C2 [×24], C4⋊D4 [×42], C22⋊Q8 [×2], C22.D4 [×20], C4.4D4 [×6], C42.C2 [×2], C42⋊2C2 [×4], C4⋊1D4 [×2], C4⋊1D4 [×10], C22×D4 [×14], C2×C4○D4 [×4], C23.36C23 [×2], C2×C4⋊1D4, C23⋊3D4 [×4], C22.29C24 [×4], C22.34C24 [×4], D42, D42 [×4], D4⋊5D4 [×4], Q8⋊6D4 [×2], C22.53C24, C22.54C24 [×4], C22.132C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×6], C25, C2×2+ (1+4) [×3], C22.132C25

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

Smallest permutation representation
►On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 26)(10 27)(11 28)(12 25)(21 32)(22 29)(23 30)(24 31)

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

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

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

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

Matrix representation ►G ⊆ 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
,
-10-200000
00110000
00100000
01-100000
00000010
00000001
00001000
00000100
,
-1-2000000
01000000
110-10000
-1-1-100000
0000-1000
00000100
00000010
0000000-1
,
10000000
01000000
00100000
00010000
00000100
0000-1000
0000000-1
00000010
,
-1-2000000
11000000
01010000
-1-1-100000
00000100
0000-1000
0000000-1
00000010
,
10000000
01000000
-10-100000
100-10000
00001000
00000100
000000-10
0000000-1

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2Q4A4B4C4D4E···4T
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim1111111111144
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C22+ (1+4)2+ (1+4)
kernelC22.132C25C23.36C23C2×C4⋊1D4C23⋊3D4C22.29C24C22.34C24D42D4⋊5D4Q8⋊6D4C22.53C24C22.54C24C4C22
# reps1214445421442

In GAP, Magma, Sage, TeX 

C_2^2._{132}C_2^5
 
% in TeX
 
G:=Group("C2^2.132C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2275);
 
// by ID
 
G=gap.SmallGroup(128,2275);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,352,2019,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=b,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^-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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