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G = C22.76C25order 128 = 27

57th central stem extension by C22 of C25

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

Aliases: C42.78C23, C22.76C25, C23.36C24, C24.502C23, (C2×Q8)⋊41D4, Q8.58(C2×D4), Q85D413C2, Q86D415C2, (C4×D4)⋊34C22, (C4×Q8)⋊36C22, C2.28(D4×C23), C41D416C22, C4⋊D480C22, C4⋊C4.483C23, (C2×C4).169C24, C4.117(C22×D4), C22⋊Q894C22, Q8(C22.D4), (C2×D4).298C23, C4.4D422C22, C22⋊C4.95C23, (C2×2- 1+4)⋊6C2, (C2×Q8).441C23, (C22×Q8)⋊30C22, C22.51(C22×D4), C22.29C2419C2, C22.19C2423C2, C22≀C2.25C22, (C23×C4).604C22, C2.15(C2.C25), (C22×C4).1207C23, (C22×D4).597C22, C23.32C2312C2, C42⋊C2.224C22, C22.D4.42C22, (C2×C4).665(C2×D4), (C22×C4○D4)⋊22C2, (C2×C4○D4)⋊24C22, SmallGroup(128,2219)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.76C25
C1C2C22C23C22×C4C23×C4C22×C4○D4 — C22.76C25
C1C22 — C22.76C25
C1C22 — C22.76C25
C1C22 — C22.76C25

Generators and relations for C22.76C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=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: 1212 in 768 conjugacy classes, 428 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×12], C4 [×14], C22, C22 [×2], C22 [×44], C2×C4 [×32], C2×C4 [×50], D4 [×68], Q8 [×16], Q8 [×12], C23, C23 [×10], C23 [×12], C42 [×12], C22⋊C4 [×36], C4⋊C4 [×20], C22×C4, C22×C4 [×27], C22×C4 [×12], C2×D4 [×46], C2×D4 [×12], C2×Q8 [×18], C2×Q8 [×16], C4○D4 [×72], C24 [×3], C42⋊C2 [×6], C4×D4 [×24], C4×Q8 [×8], C22≀C2 [×12], C4⋊D4 [×36], C22⋊Q8 [×12], C22.D4 [×4], C4.4D4 [×12], C41D4 [×12], C23×C4 [×3], C22×D4 [×3], C22×Q8, C22×Q8 [×4], C2×C4○D4 [×18], C2×C4○D4 [×8], 2- 1+4 [×8], C23.32C23, C22.19C24 [×6], C22.29C24 [×6], Q85D4 [×8], Q86D4 [×8], C22×C4○D4, C2×2- 1+4, C22.76C25
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4×C23, C2.C25 [×2], C22.76C25

Smallest permutation representation of C22.76C25
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 21)(2 22)(3 23)(4 24)(5 10)(6 11)(7 12)(8 9)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(1 28)(2 27)(3 26)(4 25)(5 17)(6 20)(7 19)(8 18)(9 31)(10 30)(11 29)(12 32)(13 24)(14 23)(15 22)(16 21)
(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 7 3 5)(2 8 4 6)(9 24 11 22)(10 21 12 23)(13 20 15 18)(14 17 16 19)(25 29 27 31)(26 30 28 32)
(1 6 3 8)(2 5 4 7)(9 21 11 23)(10 24 12 22)(13 32 15 30)(14 31 16 29)(17 25 19 27)(18 28 20 26)
(1 21)(2 22)(3 23)(4 24)(5 10)(6 11)(7 12)(8 9)(13 27)(14 28)(15 25)(16 26)(17 32)(18 29)(19 30)(20 31)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2O4A···4P4Q···4AB
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim1111111124
type+++++++++
imageC1C2C2C2C2C2C2C2D4C2.C25
kernelC22.76C25C23.32C23C22.19C24C22.29C24Q85D4Q86D4C22×C4○D4C2×2- 1+4C2×Q8C2
# reps1166881184

Matrix representation of C22.76C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
010000
100000
000010
004114
001000
000004
,
400000
040000
000200
002000
002332
004102
,
100000
040000
003000
000300
000030
000003
,
400000
040000
000100
004000
004114
003024
,
100000
010000
001000
000100
000040
003204

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

C22.76C25 in GAP, Magma, Sage, TeX

C_2^2._{76}C_2^5
% in TeX

G:=Group("C2^2.76C2^5");
// GroupNames label

G:=SmallGroup(128,2219);
// by ID

G=gap.SmallGroup(128,2219);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,184,570,136,1684]);
// Polycyclic

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