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G = C22.82C25order 128 = 27

63rd central stem extension by C22 of C25

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

Aliases: C22.82C25, C23.39C24, C42.571C23, C24.504C23, C4⋊Q832C22, (C4×D4)⋊40C22, (C4×Q8)⋊41C22, C4⋊D424C22, C4⋊C4.519C23, (C2×C4).602C24, C22⋊Q828C22, C22.32C243C2, (C2×D4).300C23, C4.4D479C22, C22⋊C4.98C23, (C2×Q8).285C23, C42.C253C22, C22.19C2428C2, C42⋊C237C22, C422C234C22, C22≀C2.27C22, (C22×C4).355C23, (C2×C42).940C22, (C23×C4).608C22, C2.19(C2.C25), C23.36C2326C2, C23.33C2318C2, C23.41C2312C2, C22.46C2414C2, C22.47C2413C2, C22.49C2411C2, C22.31C2414C2, C22.50C2418C2, C22.D4.43C22, (C2×C4⋊C4)⋊71C22, C4.138(C2×C4○D4), C4⋊C4(C22.D4), C22.15(C2×C4○D4), C2.47(C22×C4○D4), (C2×C42⋊C2)⋊66C2, (C2×C4).491(C4○D4), (C2×C4○D4).225C22, (C2×C22⋊C4).546C22, SmallGroup(128,2225)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.82C25
C1C2C22C23C22×C4C23×C4C2×C42⋊C2 — C22.82C25
C1C22 — C22.82C25
C1C22 — C22.82C25
C1C22 — C22.82C25

Generators and relations for C22.82C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=b, e2=f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=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: 732 in 520 conjugacy classes, 388 normal (18 characteristic)
C1, C2 [×3], C2 [×8], C4 [×4], C4 [×22], C22, C22 [×2], C22 [×24], C2×C4 [×2], C2×C4 [×26], C2×C4 [×30], D4 [×24], Q8 [×8], C23, C23 [×6], C23 [×4], C42 [×20], C22⋊C4 [×44], C4⋊C4 [×44], C22×C4 [×2], C22×C4 [×20], C22×C4 [×4], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×8], C24, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×22], C4×D4 [×24], C4×Q8 [×8], C22≀C2 [×4], C4⋊D4 [×20], C22⋊Q8 [×12], C22.D4 [×12], C4.4D4 [×12], C42.C2 [×8], C422C2 [×16], C4⋊Q8 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C23.33C23 [×2], C22.19C24 [×2], C23.36C23 [×4], C22.31C24, C22.32C24 [×4], C23.41C23, C22.46C24 [×4], C22.47C24 [×4], C22.49C24 [×4], C22.50C24 [×4], C22.82C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], C25, C22×C4○D4, C2.C25 [×2], C22.82C25

Smallest permutation representation of C22.82C25
On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 20)(6 17)(7 18)(8 19)(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 23)(6 9)(7 21)(8 11)(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 6 15 17)(2 7 16 18)(3 8 13 19)(4 5 14 20)(9 28 24 29)(10 25 21 30)(11 26 22 31)(12 27 23 32)
(1 17 15 6)(2 7 16 18)(3 19 13 8)(4 5 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)(5,20)(6,17)(7,18)(8,19)(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)(5,23)(6,9)(7,21)(8,11)(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), (1,6,15,17)(2,7,16,18)(3,8,13,19)(4,5,14,20)(9,28,24,29)(10,25,21,30)(11,26,22,31)(12,27,23,32), (1,17,15,6)(2,7,16,18)(3,19,13,8)(4,5,14,20)(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)(5,20)(6,17)(7,18)(8,19)(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)(5,23)(6,9)(7,21)(8,11)(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), (1,6,15,17)(2,7,16,18)(3,8,13,19)(4,5,14,20)(9,28,24,29)(10,25,21,30)(11,26,22,31)(12,27,23,32), (1,17,15,6)(2,7,16,18)(3,19,13,8)(4,5,14,20)(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),(5,20),(6,17),(7,18),(8,19),(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),(5,23),(6,9),(7,21),(8,11),(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)], [(1,6,15,17),(2,7,16,18),(3,8,13,19),(4,5,14,20),(9,28,24,29),(10,25,21,30),(11,26,22,31),(12,27,23,32)], [(1,17,15,6),(2,7,16,18),(3,19,13,8),(4,5,14,20),(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 2A2B2C2D2E2F···2K4A···4P4Q···4AF
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim11111111111124
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.82C25C2×C42⋊C2C23.33C23C22.19C24C23.36C23C22.31C24C22.32C24C23.41C23C22.46C24C22.47C24C22.49C24C22.50C24C2×C4C2
# reps11224141444484

Matrix representation of C22.82C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
003030
000001
004020
000100
,
300000
030000
003300
000200
004224
000003
,
400000
010000
002000
000200
001030
000003
,
400000
040000
003000
004200
000030
002022
,
100000
010000
004000
000400
002010
000001

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

C22.82C25 in GAP, Magma, Sage, TeX

C_2^2._{82}C_2^5
% in TeX

G:=Group("C2^2.82C2^5");
// GroupNames label

G:=SmallGroup(128,2225);
// by ID

G=gap.SmallGroup(128,2225);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184,570,136,1684]);
// Polycyclic

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