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G = C22.81C25order 128 = 27

62nd central stem extension by C22 of C25

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

Aliases: C42.81C23, C22.81C25, C24.620C23, C23.132C24, C22.12- 1+4, C22.82+ 1+4, C4⋊Q831C22, D46D419C2, D43Q818C2, (C4×D4)⋊39C22, (C2×C4).73C24, (C4×Q8)⋊40C22, C4⋊D423C22, C4⋊C4.485C23, C22⋊Q827C22, C422C23C22, (C2×D4).467C23, C22⋊C4.97C23, (C2×Q8).284C23, C42.C213C22, C42⋊C236C22, C22.19C2427C2, C22≀C2.36C22, (C22×C4).354C23, (C23×C4).607C22, C22.D45C22, C2.29(C2×2+ 1+4), C2.20(C2×2- 1+4), C22.33C243C2, C23.41C2311C2, C22.31C2413C2, C23.33C2317C2, C22.46C2413C2, C22.47C2412C2, C4⋊C4C22≀C2, (C2×C4)⋊11(C4○D4), (C22×C4⋊C4)⋊47C2, (C2×C4⋊C4)⋊70C22, C4.137(C2×C4○D4), C2.46(C22×C4○D4), C22.14(C2×C4○D4), (C2×C4○D4).224C22, SmallGroup(128,2224)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.81C25
C1C2C22C23C22×C4C23×C4C22×C4⋊C4 — C22.81C25
C1C22 — C22.81C25
C1C22 — C22.81C25
C1C22 — C22.81C25

Generators and relations for C22.81C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=g2=1, d2=b, f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=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: 812 in 562 conjugacy classes, 392 normal (18 characteristic)
C1, C2 [×3], C2 [×10], C4 [×4], C4 [×22], C22, C22 [×6], C22 [×26], C2×C4 [×28], C2×C4 [×42], D4 [×32], Q8 [×8], C23 [×3], C23 [×4], C23 [×6], C42 [×12], C22⋊C4 [×36], C4⋊C4 [×52], C22×C4 [×2], C22×C4 [×28], C22×C4 [×8], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×16], C24, C2×C4⋊C4 [×2], C2×C4⋊C4 [×14], C42⋊C2 [×10], C4×D4 [×28], C4×Q8 [×4], C22≀C2 [×4], C4⋊D4 [×16], C22⋊Q8 [×24], C22.D4 [×20], C42.C2 [×12], C422C2 [×8], C4⋊Q8 [×4], C23×C4, C23×C4 [×2], C2×C4○D4 [×6], C22×C4⋊C4, C23.33C23 [×2], C22.19C24 [×6], C22.31C24, C22.33C24 [×4], C23.41C23, D46D4 [×4], C22.46C24 [×4], C22.47C24 [×4], D43Q8 [×4], C22.81C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ 1+4 [×2], 2- 1+4 [×2], C25, C22×C4○D4, C2×2+ 1+4, C2×2- 1+4, C22.81C25

Smallest permutation representation of C22.81C25
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)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)
(1 17 15 8)(2 5 16 18)(3 19 13 6)(4 7 14 20)(9 31 24 26)(10 27 21 32)(11 29 22 28)(12 25 23 30)
(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)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim11111111111244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC22.81C25C22×C4⋊C4C23.33C23C22.19C24C22.31C24C22.33C24C23.41C23D46D4C22.46C24C22.47C24D43Q8C2×C4C22C22
# reps11261414444822

Matrix representation of C22.81C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
010000
100000
000010
000001
001000
000100
,
200000
020000
001300
000400
000042
000001
,
400000
010000
001000
000100
000010
000001
,
400000
040000
004200
004100
000042
000041
,
100000
010000
001000
000100
000040
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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,4,0,0,0,0,0,2,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C22.81C25 in GAP, Magma, Sage, TeX

C_2^2._{81}C_2^5
% in TeX

G:=Group("C2^2.81C2^5");
// GroupNames label

G:=SmallGroup(128,2224);
// by ID

G=gap.SmallGroup(128,2224);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,570,136,1684]);
// Polycyclic

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