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G = C22.84C25  order 128 = 27

65th central stem extension by C22 of C25

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

Aliases: C23.40C24, C42.82C23, C22.84C25, C24.506C23, C22.62- 1+4, C4⋊Q8⋊89C22, D4⋊3Q8⋊19C2, (C4×D4)⋊42C22, (C2×C4).75C24, (C4×Q8)⋊43C22, C4⋊C4.487C23, C22⋊Q8⋊30C22, (C2×D4).468C23, C4.4D4⋊80C22, C22⋊C4.19C23, (C2×Q8).444C23, C42.C2⋊14C22, C42⋊2C2⋊36C22, C42⋊C2⋊39C22, C22.45C24⋊5C2, C22≀C2.28C22, C4⋊D4.241C22, (C23×C4).610C22, (C2×C42).941C22, (C22×C4).357C23, C2.21(C2×2- 1+4), C2.21(C2.C25), C22.35C24⋊9C2, C22.19C24.22C2, (C22×Q8).357C22, C23.33C23⋊20C2, C23.41C23⋊14C2, C22.46C24⋊15C2, C23.36C23⋊28C2, C22.36C24⋊13C2, C22.50C24⋊19C2, C23.38C23⋊21C2, C23.37C23⋊34C2, C23.32C23⋊13C2, C22.D4.30C22, C4⋊C4○(C22⋊Q8), (C2×C4⋊C4)⋊72C22, C4.140(C2×C4○D4), (C2×C22⋊Q8)⋊76C2, C22.29(C2×C4○D4), C2.49(C22×C4○D4), (C2×C42⋊C2)⋊68C2, (C2×C4).492(C4○D4), (C2×C4○D4).226C22, (C2×C22⋊C4).381C22, SmallGroup(128,2227)

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

Derived series
C1 — C22 — C22.84C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C22.84C25
Lower central
C1 — C22 — C22.84C25
Upper central
C1 — C22 — C22.84C25
Jennings
C1 — C22 — C22.84C25

Generators and relations for C22.84C25
 G = < a,b,c,d,e,f,g | a2=b2=e2=f2=1, c2=g2=a, d2=b, ab=ba, dcd-1=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 676 in 511 conjugacy classes, 390 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C23×C4, C22×Q8, C2×C4○D4, C2×C42⋊C2, C23.32C23, C23.33C23, C2×C22⋊Q8, C22.19C24, C23.36C23, C23.37C23, C23.38C23, C22.35C24, C22.36C24, C23.41C23, C22.45C24, C22.46C24, D4⋊3Q8, C22.50C24, C22.84C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2- 1+4, C25, C22×C4○D4, C2×2- 1+4, C2.C25, C22.84C25

Smallest permutation representation of C22.84C25
►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)

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

(2 28)(4 26)(6 17)(8 19)(10 30)(12 32)(14 22)(16 24)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4N4O···4AG
order122222222224···44···4
size111122224442···24···4

44 irreducible representations

dim1111111111111111244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42- 1+4C2.C25
kernelC22.84C25C2×C42⋊C2C23.32C23C23.33C23C2×C22⋊Q8C22.19C24C23.36C23C23.37C23C23.38C23C22.35C24C22.36C24C23.41C23C22.45C24C22.46C24D4⋊3Q8C22.50C24C2×C4C22C2
# reps1111112212214822822

Matrix representation of C22.84C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
000010
000001
004000
000400
,
300000
030000
002300
000300
000032
000002
,
100000
040000
001000
000100
000040
000004
,
100000
010000
004000
003100
000040
000031
,
400000
040000
002000
000200
000030
000003

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,3,3,0,0,0,0,0,0,3,0,0,0,0,0,2,2],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;
 

C22.84C25 in GAP, Magma, Sage, TeX 

C_2^2._{84}C_2^5
 
% in TeX
 
G:=Group("C2^2.84C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2227);
 
// by ID
 
G=gap.SmallGroup(128,2227);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,456,1430,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=e^2=f^2=1,c^2=g^2=a,d^2=b,a*b=b*a,d*c*d^-1=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=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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