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G = C22.155C25  order 128 = 27

136th central stem extension by C22 of C25

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

Aliases: C23.95C24, C42.137C23, C24.159C23, C22.155C25, C4⋊Q8⋊48C22, (C4×D4)⋊76C22, (C4×Q8)⋊72C22, C4⋊D4⋊46C22, C4⋊C4.337C23, (C2×C4).145C24, (C2×C42)⋊79C22, (C2×D4).343C23, C4.4D4⋊47C22, C22⋊Q8⋊102C22, (C2×Q8).320C23, C42.C2⋊28C22, C42⋊C2⋊70C22, C42⋊2C2⋊21C22, C22.32C24⋊26C2, C22≀C2.18C22, C4⋊1D4.121C22, C22⋊C4.121C23, (C22×C4).414C23, C22.45C24⋊25C2, C22.54C24⋊14C2, C2.66(C2.C25), C22.D4⋊64C22, C22.34C24⋊28C2, C22.57C24⋊22C2, C22.47C24⋊41C2, C22.33C24⋊26C2, C22.50C24⋊42C2, C22.46C24⋊42C2, C22.56C24⋊18C2, C22.53C24⋊27C2, C22.35C24⋊27C2, C23.36C23⋊63C2, C22.49C24⋊27C2, C22.36C24⋊43C2, (C2×C4⋊C4)⋊95C22, (C2×C22⋊C4).397C22, SmallGroup(128,2298)

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

Derived series
C1 — C22 — C22.155C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C23.36C23 — C22.155C25
Lower central
C1 — C22 — C22.155C25
Upper central
C1 — C22 — C22.155C25
Jennings
C1 — C22 — C22.155C25

Generators and relations for C22.155C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=a, g2=b, ab=ba, dcd=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 692 in 477 conjugacy classes, 378 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C23.36C23, C22.32C24, C22.33C24, C22.34C24, C22.35C24, C22.36C24, C22.45C24, C22.46C24, C22.47C24, C22.49C24, C22.50C24, C22.53C24, C22.54C24, C22.56C24, C22.57C24, C22.155C25
Quotients: C1, C2, C22, C23, C24, C25, C2.C25, C22.155C25

Smallest permutation representation of C22.155C25
►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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)

(2 28)(4 26)(5 18)(6 8)(7 20)(9 29)(11 31)(13 15)(14 24)(16 22)(17 19)(21 23)

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111111111111114
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2.C25
kernelC22.155C25C23.36C23C22.32C24C22.33C24C22.34C24C22.35C24C22.36C24C22.45C24C22.46C24C22.47C24C22.49C24C22.50C24C22.53C24C22.54C24C22.56C24C22.57C24C2
# reps13331233331111216

Matrix representation of C22.155C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
44000000
00100000
40040000
00001000
00000400
00000040
00000431
,
30010000
00220000
33030000
20020000
00004314
00000030
00000200
00000431
,
20000000
02000000
00200000
00020000
00000010
00003123
00001000
00000004
,
10300000
00110000
00400000
01100000
00000100
00001000
00003123
00001043
,
43000000
01000000
04010000
01100000
00003000
00000300
00000030
00000003

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],[1,4,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,1],[3,0,3,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,2,3,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,2,4,0,0,0,0,1,3,0,3,0,0,0,0,4,0,0,1],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,3,0,4],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,1,4,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,3,1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,3,3],[4,0,0,0,0,0,0,0,3,1,4,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3] >;
 

C22.155C25 in GAP, Magma, Sage, TeX 

C_2^2._{155}C_2^5
 
% in TeX
 
G:=Group("C2^2.155C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2298);
 
// by ID
 
G=gap.SmallGroup(128,2298);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,184,2019,570,1684,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=a,g^2=b,a*b=b*a,d*c*d=g*c*g^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f=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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