Copied to
clipboard

G = C22.90C25order 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, D48(C2×Q8), (C2×D4)⋊23Q8, (D4×Q8)⋊19C2, C232(C2×Q8), D42(C22⋊Q8), C4⋊Q892C22, D43Q820C2, (C2×C4).80C24, (C4×Q8)⋊44C22, C232Q86C2, C2.15(Q8×C23), C4.53(C22×Q8), C4⋊C4.296C23, C22⋊Q833C22, (C2×D4).505C23, (C4×D4).232C22, (C2×Q8).287C23, C42.C215C22, (C22×Q8)⋊33C22, C22.11(C22×Q8), C22⋊C4.100C23, (C22×C4).362C23, (C23×C4).611C22, (C2×C42).944C22, C2.33(C2×2+ 1+4), C2.25(C2.C25), C22.11C24.10C2, (C22×D4).599C22, C23.37C2335C2, C42⋊C2.225C22, C23.41C2315C2, (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

C1C22 — C22.90C25
C1C2C22C23C24C23×C4C2×C4×D4 — C22.90C25
C1C22 — C22.90C25
C1C22 — C22.90C25
C1C22 — C22.90C25

Generators and relations for C22.90C25
 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 >

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], C232Q8 [×4], C23.41C23 [×2], D4×Q8 [×4], D43Q8 [×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

Smallest permutation representation of C22.90C25
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 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 23)(2 32)(3 21)(4 30)(5 14)(6 11)(7 16)(8 9)(10 20)(12 18)(13 19)(15 17)(22 26)(24 28)(25 29)(27 31)
(1 9 3 11)(2 12 4 10)(5 24 7 22)(6 23 8 21)(13 25 15 27)(14 28 16 26)(17 31 19 29)(18 30 20 32)
(5 20)(6 17)(7 18)(8 19)(21 29)(22 30)(23 31)(24 32)
(1 3)(2 26)(4 28)(5 18)(6 8)(7 20)(9 11)(10 16)(12 14)(13 15)(17 19)(21 23)(22 32)(24 30)(25 27)(29 31)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim111111111244
type+++++++++-+
imageC1C2C2C2C2C2C2C2C2Q82+ 1+4C2.C25
kernelC22.90C25C2×C4×D4C22.11C24C2×C22⋊Q8C23.37C23C232Q8C23.41C23D4×Q8D43Q8C2×D4C4C2
# reps1124242412822

Matrix representation of C22.90C25 in 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] >;

C22.90C25 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

׿
×
𝔽