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G = C22.89C25order 128 = 27

70th central stem extension by C22 of C25

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

Aliases: C22.89C25, C23.45C24, C42.577C23, C24.136C23, C4○D412D4, D4.60(C2×D4), D4(C4.4D4), C4⋊Q891C22, Q8.62(C2×D4), Q8(C4.4D4), D45D419C2, Q85D417C2, (C4×D4)⋊45C22, (C2×C4).79C24, C2.35(D4×C23), C22≀C29C22, C4⋊C4.295C23, C41D452C22, C4⋊D428C22, (C2×C42)⋊60C22, (C4×Q8)⋊100C22, C4.124(C22×D4), C22⋊Q832C22, (C2×D4).472C23, C4.4D483C22, C22⋊C4.24C23, (C2×Q8).449C23, (C22×Q8)⋊32C22, C22.16(C22×D4), C22.29C2423C2, (C22×C4).361C23, (C2×2+ 1+4)⋊12C2, (C2×2- 1+4)⋊10C2, C22.D47C22, C42⋊C2101C22, C2.24(C2.C25), C22.26C2438C2, (C22×D4).426C22, C23.38C2323C2, (C4×C4○D4)⋊28C2, (C2×C4).189(C2×D4), (C2×C4.4D4)⋊54C2, (C2×C4○D4)⋊31C22, (C2×C22⋊C4)⋊49C22, SmallGroup(128,2232)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.89C25
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C22.89C25
C1C22 — C22.89C25
C1C22 — C22.89C25
C1C22 — C22.89C25

Generators and relations for C22.89C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=ba=ab, g2=a, dcd=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: 1260 in 770 conjugacy classes, 428 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×8], C4 [×16], C22, C22 [×6], C22 [×42], C2×C4, C2×C4 [×31], C2×C4 [×42], D4 [×12], D4 [×56], Q8 [×4], Q8 [×20], C23 [×11], C23 [×24], C42, C42 [×9], C22⋊C4 [×46], C4⋊C4 [×18], C22×C4 [×27], C2×D4, C2×D4 [×36], C2×D4 [×36], C2×Q8 [×2], C2×Q8 [×9], C2×Q8 [×20], C4○D4 [×8], C4○D4 [×56], C24 [×6], C2×C42 [×3], C2×C22⋊C4 [×12], C42⋊C2 [×3], C4×D4 [×18], C4×Q8 [×2], C22≀C2 [×12], C4⋊D4 [×24], C22⋊Q8 [×12], C22.D4 [×12], C4.4D4, C4.4D4 [×21], C41D4 [×3], C4⋊Q8 [×3], C22×D4 [×9], C22×Q8 [×5], C2×C4○D4, C2×C4○D4 [×14], 2+ 1+4 [×8], 2- 1+4 [×8], C4×C4○D4, C2×C4.4D4 [×3], C22.26C24 [×3], C22.29C24 [×3], C23.38C23 [×3], D45D4 [×12], Q85D4 [×4], C2×2+ 1+4, C2×2- 1+4, C22.89C25
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4×C23, C2.C25 [×2], C22.89C25

Smallest permutation representation of C22.89C25
On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(1 7)(2 8)(3 5)(4 6)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)
(1 15)(2 24)(3 13)(4 22)(5 21)(6 14)(7 23)(8 16)(9 25)(10 18)(11 27)(12 20)(17 29)(19 31)(26 30)(28 32)
(1 19)(2 20)(3 17)(4 18)(5 25)(6 26)(7 27)(8 28)(9 15)(10 16)(11 13)(12 14)(21 31)(22 32)(23 29)(24 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)(2 32)(3 29)(4 30)(5 9)(6 10)(7 11)(8 12)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)
(1 21 5 15)(2 22 6 16)(3 23 7 13)(4 24 8 14)(9 19 31 25)(10 20 32 26)(11 17 29 27)(12 18 30 28)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2Q4A···4L4M···4Z
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C2.C25
kernelC22.89C25C4×C4○D4C2×C4.4D4C22.26C24C22.29C24C23.38C23D45D4Q85D4C2×2+ 1+4C2×2- 1+4C4○D4C2
# reps1133331241184

Matrix representation of C22.89C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
010000
000100
001000
000001
000010
,
100000
010000
000001
000040
000400
001000
,
040000
100000
002000
000200
000020
000002
,
100000
010000
000010
000001
001000
000100
,
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],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[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] >;

C22.89C25 in GAP, Magma, Sage, TeX

C_2^2._{89}C_2^5
% in TeX

G:=Group("C2^2.89C2^5");
// GroupNames label

G:=SmallGroup(128,2232);
// by ID

G=gap.SmallGroup(128,2232);
# by ID

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

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