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G = C22.99C25order 128 = 27

80th central stem extension by C22 of C25

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

Aliases: C42.91C23, C22.99C25, C23.140C24, C24.139C23, C4⋊Q834C22, (C4×D4)⋊49C22, (C2×C4).89C24, (C4×Q8)⋊48C22, C4⋊C4.494C23, (C2×C42)⋊63C22, C22⋊Q835C22, (C2×D4).306C23, C4.4D428C22, (C2×Q8).291C23, C42.C275C22, C22.45C247C2, C422C237C22, C22.11C2420C2, C42⋊C242C22, C22≀C2.10C22, C4⋊D4.226C22, C41D4.113C22, C22⋊C4.108C23, (C22×C4).369C23, C2.31(C2.C25), C22.29C24.16C2, (C22×D4).428C22, C22.D453C22, (C22×Q8).362C22, C22.50C2423C2, C23.37C2342C2, C23.32C2315C2, C22.36C2414C2, C23.36C2331C2, C22.53C2414C2, C23.38C2325C2, (C4×C4○D4)⋊30C2, C4.182(C2×C4○D4), (C2×C4.4D4)⋊55C2, C22.32(C2×C4○D4), C2.55(C22×C4○D4), (C2×C4).308(C4○D4), (C2×C4○D4).331C22, (C2×C22⋊C4).384C22, SmallGroup(128,2242)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.99C25
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C22.99C25
C1C22 — C22.99C25
C1C22 — C22.99C25
C1C22 — C22.99C25

Generators and relations for C22.99C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=b, e2=ba=ab, g2=a, dcd-1=gcg-1=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: 748 in 520 conjugacy classes, 388 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×22], C22, C22 [×2], C22 [×24], C2×C4 [×2], C2×C4 [×26], C2×C4 [×22], D4 [×22], Q8 [×14], C23, C23 [×6], C23 [×8], C42 [×2], C42 [×20], C22⋊C4 [×50], C4⋊C4 [×38], C22×C4, C22×C4 [×18], C2×D4, C2×D4 [×14], C2×D4 [×4], C2×Q8, C2×Q8 [×8], C2×Q8 [×4], C4○D4 [×4], C24 [×2], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×8], C42⋊C2 [×3], C42⋊C2 [×14], C4×D4 [×22], C4×Q8 [×14], C22≀C2 [×4], C4⋊D4 [×4], C22⋊Q8 [×16], C22.D4 [×20], C4.4D4 [×20], C42.C2 [×2], C422C2 [×16], C41D4 [×2], C4⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C4×C4○D4, C22.11C24, C23.32C23, C2×C4.4D4, C23.36C23 [×4], C23.37C23, C22.29C24, C23.38C23, C22.36C24 [×4], C22.45C24 [×8], C22.50C24 [×4], C22.53C24 [×4], C22.99C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], C25, C22×C4○D4, C2.C25 [×2], C22.99C25

Smallest permutation representation of C22.99C25
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 11)(2 16)(3 9)(4 14)(5 32)(6 21)(7 30)(8 23)(10 26)(12 28)(13 25)(15 27)(17 29)(18 22)(19 31)(20 24)
(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 29 25 23)(2 30 26 24)(3 31 27 21)(4 32 28 22)(5 10 18 16)(6 11 19 13)(7 12 20 14)(8 9 17 15)
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)
(1 15 27 11)(2 16 28 12)(3 13 25 9)(4 14 26 10)(5 22 20 30)(6 23 17 31)(7 24 18 32)(8 21 19 29)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A···4P4Q···4AF
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim111111111111124
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.99C25C4×C4○D4C22.11C24C23.32C23C2×C4.4D4C23.36C23C23.37C23C22.29C24C23.38C23C22.36C24C22.45C24C22.50C24C22.53C24C2×C4C2
# reps111114111484484

Matrix representation of C22.99C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
240000
000100
001000
000404
004040
,
300000
030000
002040
000204
003030
000303
,
320000
020000
002000
000200
000020
000002
,
400000
040000
001000
000100
004040
000404
,
400000
040000
000100
004000
000001
000040

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

C22.99C25 in GAP, Magma, Sage, TeX

C_2^2._{99}C_2^5
% in TeX

G:=Group("C2^2.99C2^5");
// GroupNames label

G:=SmallGroup(128,2242);
// by ID

G=gap.SmallGroup(128,2242);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,520,570,1684,102]);
// Polycyclic

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