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G = C22.89C25  order 128 = 27

70th central stem extension by C22 of C25

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

Aliases: C23.45C24, C22.89C25, C42.577C23, C24.136C23, C4○D4⋊12D4, D4.60(C2×D4), D4○(C4.4D4), C4⋊Q8⋊91C22, Q8.62(C2×D4), Q8○(C4.4D4), D4⋊5D4⋊19C2, Q8⋊5D4⋊17C2, (C4×D4)⋊45C22, (C2×C4).79C24, C2.35(D4×C23), C22≀C2⋊9C22, C4⋊1D4⋊52C22, C4⋊D4⋊28C22, C4⋊C4.295C23, (C2×C42)⋊60C22, (C4×Q8)⋊100C22, C4.124(C22×D4), C22⋊Q8⋊32C22, (C2×D4).472C23, C4.4D4⋊83C22, C22⋊C4.24C23, (C2×Q8).449C23, (C22×Q8)⋊32C22, C22.16(C22×D4), C22.29C24⋊23C2, (C22×C4).361C23, (C2×2- (1+4))⋊10C2, (C2×2+ (1+4))⋊12C2, C22.D4⋊7C22, C42⋊C2⋊101C22, C2.24(C2.C25), C22.26C24⋊38C2, (C22×D4).426C22, C23.38C23⋊23C2, (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

Derived series
C1 — C22 — C22.89C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.89C25
Lower central
C1 — C22 — C22.89C25
Upper central
C1 — C22 — C22.89C25
Jennings
C1 — C22 — C22.89C25

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], C4⋊1D4 [×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], D4⋊5D4 [×12], Q8⋊5D4 [×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

Generators and relations
 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 >

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ 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] >;
 

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×C4.4D4C22.26C24C22.29C24C23.38C23D4⋊5D4Q8⋊5D4C2×2+ (1+4)C2×2- (1+4)C4○D4C2
# reps1133331241184

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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