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G = C22.130C25  order 128 = 27

111st central stem extension by C22 of C25

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

Aliases: C23.71C24, C42.113C23, C22.130C25, C24.518C23, C4.892+ (1+4), C22.32- (1+4), (D4×Q8)⋊27C2, C4⋊Q8⋊43C22, D4⋊6D4⋊35C2, Q8⋊5D4⋊28C2, D4⋊3Q8⋊34C2, (C4×D4)⋊64C22, (C4×Q8)⋊61C22, C4⋊C4.318C23, C4⋊D4⋊38C22, (C2×C4).120C24, C22⋊Q8⋊48C22, (C2×D4).322C23, C4.4D4⋊39C22, C22⋊C4.46C23, (C2×Q8).463C23, C42.C2⋊23C22, (C22×Q8)⋊42C22, C22.19C24⋊43C2, C22.32C24⋊17C2, C42⋊C2⋊58C22, C42⋊2C2⋊17C22, C22≀C2.33C22, (C23×C4).620C22, (C22×C4).390C23, C2.59(C2×2+ (1+4)), C2.42(C2×2- (1+4)), C2.48(C2.C25), C22.56C24⋊6C2, C22.57C24⋊7C2, C22.D4⋊19C22, C22.46C24⋊31C2, C22.36C24⋊28C2, C22.35C24⋊18C2, C22.31C24⋊22C2, C22.50C24⋊32C2, C22.33C24⋊16C2, C22.47C24⋊30C2, C23.38C23⋊29C2, (C2×C4⋊C4)⋊88C22, (C2×C22⋊Q8)⋊85C2, (C2×C4○D4)⋊47C22, (C2×C22⋊C4).392C22, SmallGroup(128,2273)

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

Derived series
C1 — C22 — C22.130C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.130C25
Lower central
C1 — C22 — C22.130C25
Upper central
C1 — C22 — C22.130C25
Jennings
C1 — C22 — C22.130C25

Subgroups: 772 in 516 conjugacy classes, 382 normal (122 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×23], C22, C22 [×2], C22 [×24], C2×C4 [×10], C2×C4 [×14], C2×C4 [×27], D4 [×23], Q8 [×13], C23 [×5], C23 [×2], C23 [×4], C42 [×2], C42 [×10], C22⋊C4 [×10], C22⋊C4 [×34], C4⋊C4 [×10], C4⋊C4 [×38], C22×C4 [×8], C22×C4 [×14], C22×C4 [×2], C2×D4 [×8], C2×D4 [×10], C2×Q8 [×4], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], C24, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×3], C42⋊C2 [×2], C4×D4 [×5], C4×D4 [×14], C4×Q8 [×3], C4×Q8 [×2], C22≀C2 [×2], C22≀C2 [×2], C4⋊D4 [×5], C4⋊D4 [×14], C22⋊Q8 [×11], C22⋊Q8 [×20], C22.D4 [×2], C22.D4 [×16], C4.4D4, C4.4D4 [×10], C42.C2, C42.C2 [×8], C42⋊2C2 [×14], C4⋊Q8 [×2], C4⋊Q8 [×4], C23×C4, C22×Q8 [×2], C2×C4○D4 [×3], C2×C22⋊Q8, C22.19C24 [×2], C23.38C23, C22.31C24, C22.32C24 [×4], C22.33C24 [×2], C22.35C24, C22.36C24, C22.36C24 [×2], D4⋊6D4, Q8⋊5D4 [×2], D4×Q8, C22.46C24, C22.47C24, C22.47C24 [×2], D4⋊3Q8, D4⋊3Q8 [×2], C22.50C24, C22.56C24 [×2], C22.57C24 [×2], C22.130C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×2], 2- (1+4) [×2], C25, C2×2+ (1+4), C2×2- (1+4), C2.C25, C22.130C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, 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 9)(2 10)(3 11)(4 12)(5 31)(6 32)(7 29)(8 30)(13 19)(14 20)(15 17)(16 18)(21 27)(22 28)(23 25)(24 26)

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10300000
00110000
00400000
01100000
00000034
00000002
00002100
00000300
,
43000000
01000000
04010000
01100000
00000010
00000001
00001000
00000100
,
40000000
11000000
40100000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
10400000
40040000
00004200
00004100
00000013
00000014
,
10000000
01000000
00100000
00010000
00001300
00001400
00000013
00000014

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4Z
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim111111111111111111444
type+++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ (1+4)2- (1+4)C2.C25
kernelC22.130C25C2×C22⋊Q8C22.19C24C23.38C23C22.31C24C22.32C24C22.33C24C22.35C24C22.36C24D4⋊6D4Q8⋊5D4D4×Q8C22.46C24C22.47C24D4⋊3Q8C22.50C24C22.56C24C22.57C24C4C22C2
# reps112114213121133122222

In GAP, Magma, Sage, TeX 

C_2^2._{130}C_2^5
 
% in TeX
 
G:=Group("C2^2.130C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2273);
 
// by ID
 
G=gap.SmallGroup(128,2273);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=1,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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