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G = C22.157C25  order 128 = 27

138th central stem extension by C22 of C25

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

Aliases: C23.97C24, C24.160C23, C22.157C25, C42.139C23, C4⋊Q8⋊49C22, (C4×D4)⋊77C22, (C4×Q8)⋊73C22, C4⋊C4.510C23, (C2×C4).147C24, (C2×C42)⋊80C22, C22⋊Q8⋊58C22, C22≀C2⋊22C22, C24⋊C22⋊9C2, (C2×D4).345C23, C4.4D4⋊48C22, C22⋊C4.65C23, (C2×Q8).322C23, C42.C2⋊71C22, C42⋊2C2⋊50C22, C22.32C24⋊27C2, C42⋊C2⋊71C22, C4⋊D4.127C22, (C22×C4).416C23, C22.45C24⋊26C2, C2.68(C2.C25), C22.D4⋊65C22, C22.50C24⋊43C2, C23.36C23⋊65C2, C22.36C24⋊44C2, C22.57C24⋊23C2, (C2×C22⋊C4)⋊67C22, SmallGroup(128,2300)

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

Derived series
C1 — C22 — C22.157C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.36C23 — C22.157C25
Lower central
C1 — C22 — C22.157C25
Upper central
C1 — C22 — C22.157C25
Jennings
C1 — C22 — C22.157C25

Subgroups: 708 in 479 conjugacy classes, 378 normal (7 characteristic)
C1, C2 [×3], C2 [×7], C4 [×24], C22, C22 [×25], C2×C4 [×24], C2×C4 [×15], D4 [×15], Q8 [×9], C23, C23 [×6], C23 [×6], C42, C42 [×17], C22⋊C4 [×54], C4⋊C4 [×42], C22×C4 [×15], C2×D4 [×15], C2×Q8 [×9], C24 [×2], C2×C42, C2×C22⋊C4 [×6], C42⋊C2 [×12], C4×D4 [×15], C4×Q8 [×9], C22≀C2 [×8], C4⋊D4 [×9], C22⋊Q8 [×21], C22.D4 [×18], C4.4D4 [×21], C42.C2 [×3], C42⋊2C2 [×26], C4⋊Q8 [×6], C23.36C23 [×3], C22.32C24 [×6], C22.36C24 [×6], C22.45C24 [×6], C22.50C24 [×6], C24⋊C22, C22.57C24 [×3], C22.157C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], C25, C2.C25 [×3], C22.157C25

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
31000000
02000000
02020000
03200000
00000020
00002224
00003000
00000333
,
24000000
33000000
33020000
02300000
00000020
00003331
00003000
00002202
,
30000000
03000000
00300000
00030000
00000010
00004443
00001000
00000001
,
40300000
00110000
10100000
44400000
00000100
00004000
00004443
00001011
,
43000000
11000000
01010000
44400000
00002000
00000200
00000020
00000002

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

38 conjugacy classes

class 1 2A2B2C2D···2J4A···4F4G···4AA
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim111111114
type++++++++
imageC1C2C2C2C2C2C2C2C2.C25
kernelC22.157C25C23.36C23C22.32C24C22.36C24C22.45C24C22.50C24C24⋊C22C22.57C24C2
# reps136666136

In GAP, Magma, Sage, TeX 

C_2^2._{157}C_2^5
 
% in TeX
 
G:=Group("C2^2.157C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2300);
 
// by ID
 
G=gap.SmallGroup(128,2300);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,723,184,2019,570,360,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=1,c^2=e^2=a,f^2=g^2=b*a=a*b,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^-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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