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G = C22.90C25  order 128 = 27

71st central stem extension by C22 of C25

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

Aliases: C22.90C25, C23.134C24, C42.578C23, C24.507C23, C4.792+ (1+4), D4⋊8(C2×Q8), (D4×Q8)⋊19C2, (C2×D4)⋊23Q8, C23⋊2(C2×Q8), D4○2(C22⋊Q8), C4⋊Q8⋊92C22, D4⋊3Q8⋊20C2, (C2×C4).80C24, (C4×Q8)⋊44C22, C23⋊2Q8⋊6C2, C4.53(C22×Q8), C2.15(Q8×C23), C4⋊C4.296C23, C22⋊Q8⋊33C22, (C2×D4).505C23, (C4×D4).232C22, (C2×Q8).287C23, C42.C2⋊15C22, (C22×Q8)⋊33C22, C22.11(C22×Q8), C22⋊C4.100C23, (C2×C42).944C22, (C23×C4).611C22, (C22×C4).362C23, C2.33(C2×2+ (1+4)), C2.25(C2.C25), C22.11C24.10C2, (C22×D4).599C22, C23.37C23⋊35C2, C42⋊C2.225C22, C23.41C23⋊15C2, (C2×C4)⋊3(C2×Q8), (C2×C4×D4).90C2, (C2×D4)○(C22⋊Q8), (C2×C4⋊C4)⋊74C22, (C2×C22⋊Q8)⋊77C2, (C2×C22⋊C4).382C22, SmallGroup(128,2233)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C22.90C25
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C4×D4 — C22.90C25
Lower central
C1 — C22 — C22.90C25
Upper central
C1 — C22 — C22.90C25
Jennings
C1 — C22 — C22.90C25

Subgroups: 788 in 550 conjugacy classes, 430 normal (16 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×23], C22, C22 [×10], C22 [×18], C2×C4 [×2], C2×C4 [×28], C2×C4 [×33], D4 [×16], Q8 [×20], C23, C23 [×12], C23 [×4], C42 [×12], C22⋊C4 [×32], C4⋊C4 [×60], C22×C4 [×3], C22×C4 [×26], C22×C4 [×4], C2×D4 [×12], C2×Q8 [×16], C2×Q8 [×8], C24 [×2], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×12], C42⋊C2 [×4], C4×D4 [×24], C4×Q8 [×8], C22⋊Q8 [×64], C42.C2 [×12], C4⋊Q8 [×12], C23×C4 [×2], C22×D4, C22×Q8 [×4], C2×C4×D4, C22.11C24 [×2], C2×C22⋊Q8 [×4], C23.37C23 [×2], C23⋊2Q8 [×4], C23.41C23 [×2], D4×Q8 [×4], D4⋊3Q8 [×12], C22.90C25

Quotients:
C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C24 [×31], C22×Q8 [×14], 2+ (1+4) [×2], C25, Q8×C23, C2×2+ (1+4), C2.C25, C22.90C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=d2=f2=g2=1, c2=e2=b, ab=ba, dcd=gcg=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 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)

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

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

(5 30)(6 31)(7 32)(8 29)(17 23)(18 24)(19 21)(20 22)

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
030000
300000
000010
000001
001000
000100
,
400000
040000
000100
001000
000004
000040
,
040000
100000
004000
000400
000040
000004
,
400000
040000
001000
000400
000010
000004
,
100000
010000
001000
000100
000040
000004

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

44 conjugacy classes

class 1 2A2B2C2D···2M4A···4H4I···4AD
order12222···24···44···4
size11112···22···24···4

44 irreducible representations

dim111111111244
type+++++++++-+
imageC1C2C2C2C2C2C2C2C2Q82+ (1+4)C2.C25
kernelC22.90C25C2×C4×D4C22.11C24C2×C22⋊Q8C23.37C23C23⋊2Q8C23.41C23D4×Q8D4⋊3Q8C2×D4C4C2
# reps1124242412822

In GAP, Magma, Sage, TeX 

C_2^2._{90}C_2^5
 
% in TeX
 
G:=Group("C2^2.90C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2233);
 
// by ID
 
G=gap.SmallGroup(128,2233);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,352,570,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=f^2=g^2=1,c^2=e^2=b,a*b=b*a,d*c*d=g*c*g=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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