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G = C22.80C25order 128 = 27

61st central stem extension by C22 of C25

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

Aliases: C22.80C25, C23.38C24, C42.570C23, C24.133C23, (C4×D4)⋊38C22, (C4×Q8)⋊39C22, C232Q85C2, C4⋊C4.484C23, (C2×C4).170C24, (C2×C42)⋊56C22, C233D4.8C2, C22⋊Q826C22, C422C22C22, C22≀C2.7C22, (C2×D4).466C23, C4.4D478C22, C22⋊C4.96C23, (C2×Q8).283C23, C42.C252C22, C42⋊C235C22, C22.45C243C2, C22.11C2416C2, C23.198(C4○D4), C4⋊D4.224C22, (C23×C4).606C22, (C22×C4).353C23, C22.D44C22, C2.18(C2.C25), C22.33C242C2, (C22×D4).424C22, C22.46C2412C2, C23.36C2325C2, (C2×C4⋊C4)⋊141C22, C22.27(C2×C4○D4), C2.45(C22×C4○D4), (C2×C42⋊C2)⋊65C2, (C2×C22⋊C4)⋊47C22, (C2×C22.D4)⋊60C2, C22⋊C4(C22.D4), SmallGroup(128,2223)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.80C25
C1C2C22C23C24C23×C4C2×C42⋊C2 — C22.80C25
C1C22 — C22.80C25
C1C22 — C22.80C25
C1C22 — C22.80C25

Generators and relations for C22.80C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=b, e2=a, ab=ba, dcd-1=gcg=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 764 in 526 conjugacy classes, 388 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×24], C22, C22 [×6], C22 [×26], C2×C4 [×24], C2×C4 [×32], D4 [×20], Q8 [×4], C23, C23 [×10], C23 [×10], C42 [×16], C22⋊C4 [×48], C4⋊C4 [×48], C22×C4 [×2], C22×C4 [×22], C22×C4 [×4], C2×D4 [×12], C2×D4 [×8], C2×Q8 [×4], C24, C24 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×6], C42⋊C2 [×20], C4×D4 [×20], C4×Q8 [×4], C22≀C2 [×4], C4⋊D4 [×4], C22⋊Q8 [×20], C22.D4 [×28], C4.4D4 [×4], C42.C2 [×12], C422C2 [×16], C23×C4, C22×D4 [×2], C2×C42⋊C2, C22.11C24 [×2], C2×C22.D4 [×2], C23.36C23 [×4], C233D4, C22.33C24 [×4], C232Q8, C22.45C24 [×8], C22.46C24 [×8], C22.80C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], C25, C22×C4○D4, C2.C25 [×2], C22.80C25

Smallest permutation representation of C22.80C25
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 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 22)(2 31)(3 24)(4 29)(5 15)(6 12)(7 13)(8 10)(9 18)(11 20)(14 19)(16 17)(21 26)(23 28)(25 32)(27 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 31 27 23)(2 32 28 24)(3 29 25 21)(4 30 26 22)(5 10 20 14)(6 11 17 15)(7 12 18 16)(8 9 19 13)
(1 11)(2 16)(3 9)(4 14)(5 30)(6 23)(7 32)(8 21)(10 26)(12 28)(13 25)(15 27)(17 31)(18 24)(19 29)(20 22)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4L4M···4AD
order12222···222224···44···4
size11112···244442···24···4

44 irreducible representations

dim111111111124
type++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.80C25C2×C42⋊C2C22.11C24C2×C22.D4C23.36C23C233D4C22.33C24C232Q8C22.45C24C22.46C24C23C2
# reps112241418884

Matrix representation of C22.80C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
030000
200000
000020
001431
003000
002011
,
300000
030000
000010
002312
001000
004102
,
010000
100000
002000
000200
000020
000002
,
400000
040000
000100
001000
003243
002301
,
100000
010000
000100
001000
002312
000004

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

C22.80C25 in GAP, Magma, Sage, TeX

C_2^2._{80}C_2^5
% in TeX

G:=Group("C2^2.80C2^5");
// GroupNames label

G:=SmallGroup(128,2223);
// by ID

G=gap.SmallGroup(128,2223);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184,570,1684]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=b,e^2=a,a*b=b*a,d*c*d^-1=g*c*g=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=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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