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G = C22.44C25  order 128 = 27

25th central stem extension by C22 of C25

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

Aliases: C22.44C25, C23.21C24, C24.486C23, C42.546C23, C4⋊Q8⋊77C22, (C4×D4)⋊97C22, (C4×Q8)⋊31C22, C4⋊C4.285C23, C4⋊1D4⋊45C22, (C2×C4).164C24, (C2×C42)⋊47C22, C22⋊Q8⋊78C22, C4○(C22.32C24), (C2×D4).293C23, C4.4D4⋊67C22, C22⋊C4.11C23, (C2×Q8).426C23, C42.C2⋊42C22, C42⋊2C2⋊25C22, C22.32C24⋊28C2, C22.19C24⋊14C2, C42⋊C2⋊91C22, C22≀C2.21C22, C4⋊D4.218C22, C2.5(C2.C25), (C23×C4).588C22, C4○(C22.35C24), C4○(C22.34C24), C4○(C22.33C24), C4○(C22.36C24), (C22×C4).1184C23, C22.26C24⋊27C2, C22.D4⋊37C22, C22.36C24⋊45C2, C22.35C24⋊28C2, C22.34C24⋊30C2, C23.37C23⋊28C2, C22.33C24⋊28C2, C23.36C23⋊17C2, (C4×C4○D4)⋊17C2, C4.75(C2×C4○D4), C22.10(C2×C4○D4), C2.21(C22×C4○D4), (C2×C42⋊C2)⋊58C2, (C2×C4).718(C4○D4), (C2×C4⋊C4).950C22, (C2×C4)○(C22.32C24), (C2×C4○D4).322C22, (C2×C22⋊C4).533C22, (C2×C4)○(C22.35C24), (C2×C4)○(C22.36C24), SmallGroup(128,2187)

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

Derived series
C1 — C22 — C22.44C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C22.44C25
Lower central
C1 — C22 — C22.44C25
Upper central
C1 — C2×C4 — C22.44C25
Jennings
C1 — C22 — C22.44C25

Generators and relations for C22.44C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=b, g2=a, ab=ba, dcd-1=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 732 in 520 conjugacy classes, 388 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C42, 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, C4⋊Q8, C23×C4, C2×C4○D4, C2×C42⋊C2, C4×C4○D4, C22.19C24, C23.36C23, C22.26C24, C23.37C23, C22.32C24, C22.33C24, C22.34C24, C22.35C24, C22.36C24, C22.44C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, C25, C22×C4○D4, C2.C25, C22.44C25

Smallest permutation representation of C22.44C25
►On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 24)(10 21)(11 22)(12 23)(25 30)(26 31)(27 32)(28 29)

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

(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)

(2 16)(4 14)(5 18)(7 20)(9 11)(10 23)(12 21)(22 24)(25 32)(26 28)(27 30)(29 31)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4N4O···4AF
order1222222···244444···44···4
size1111224···411112···24···4

44 irreducible representations

dim11111111111124
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.44C25C2×C42⋊C2C4×C4○D4C22.19C24C23.36C23C22.26C24C23.37C23C22.32C24C22.33C24C22.34C24C22.35C24C22.36C24C2×C4C2
# reps11228114422484

Matrix representation of C22.44C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
040000
400000
000010
002313
001000
001402
,
300000
030000
000100
001000
003242
003201
,
100000
040000
001000
000400
000010
002014
,
100000
010000
001000
000100
000040
002304
,
100000
010000
003000
000300
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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,2,1,1,0,0,0,3,0,4,0,0,1,1,0,0,0,0,0,3,0,2],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,3,3,0,0,1,0,2,2,0,0,0,0,4,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,2,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,2,0,0,0,1,0,3,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;
 

C22.44C25 in GAP, Magma, Sage, TeX 

C_2^2._{44}C_2^5
 
% in TeX
 
G:=Group("C2^2.44C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2187);
 
// by ID
 
G=gap.SmallGroup(128,2187);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,387,1123,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b,g^2=a,a*b=b*a,d*c*d^-1=f*c*f=a*c=c*a,e*d*e=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*g=g*c,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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