Copied to
clipboard

G = C4.C25  order 128 = 27

13rd non-split extension by C4 of C25 acting via C25/C24=C2

p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: C8.8C24, C4.13C25, D8.9C23, D4.10C24, Q8.10C24, Q16.8C23, SD16.2C23, M4(2).21C23, 2- 1+4⋊11C22, 2+ 1+4⋊12C22, Q8○D8⋊7C2, Q8○(C8⋊C22), D4○SD16⋊6C2, C4○D4.55D4, D4.66(C2×D4), C4○D8⋊4C22, C8○D4⋊8C22, Q8.68(C2×D4), D4○(C8.C22), (C2×D4).338D4, Q8○M4(2)⋊4C2, (C2×Q8).257D4, C2.48(D4×C23), C8⋊C22⋊16C22, (C2×C4).150C24, (C2×C8).125C23, C2.C25⋊7C2, (C2×Q16)⋊35C22, C4○D4.18C23, C4.130(C22×D4), C23.359(C2×D4), D8⋊C22⋊12C2, (C2×SD16)⋊37C22, (C2×D4).348C23, C8.C22⋊16C22, (C2×Q8).325C23, (C22×Q8)⋊49C22, C22.22(C22×D4), (C2×M4(2))⋊35C22, (C22×C4).418C23, (C2×2- 1+4)⋊13C2, (C2×C4).671(C2×D4), (C2×C4○D4)⋊60C22, (C2×C8.C22)⋊36C2, SmallGroup(128,2318)

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

Derived series
C1 — C4 — C4.C25
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — C4.C25
Lower central
C1 — C2 — C4 — C4.C25
Upper central
C1 — C2 — C2×C4○D4 — C4.C25
Jennings
C1 — C2 — C2 — C4 — C4.C25

Generators and relations for C4.C25
 G = < a,b,c,d,e,f | a4=b2=d2=e2=f2=1, c2=a2, bab=cac-1=a-1, ad=da, ae=ea, af=fa, cbc-1=ab, bd=db, be=eb, fbf=a2b, cd=dc, ce=ec, cf=fc, ede=a2d, df=fd, ef=fe >

Subgroups: 1044 in 706 conjugacy classes, 426 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C2×M4(2), C8○D4, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C8.C22, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, Q8○M4(2), C2×C8.C22, D8⋊C22, D4○SD16, Q8○D8, C2×2- 1+4, C2.C25, C4.C25
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D4×C23, C4.C25

Smallest permutation representation of C4.C25
►On 32 points
Generators in S32
(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 11)(2 10)(3 9)(4 12)(5 15)(6 14)(7 13)(8 16)(17 26)(18 25)(19 28)(20 27)(21 32)(22 31)(23 30)(24 29)

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

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

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

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

41 conjugacy classes

class 1 2A2B···2H2I···2N4A···4H4I···4R8A···8H
order122···22···24···44···48···8
size112···24···42···24···44···4

41 irreducible representations

dim111111112228
type+++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4C4.C25
kernelC4.C25Q8○M4(2)C2×C8.C22D8⋊C22D4○SD16Q8○D8C2×2- 1+4C2.C25C2×D4C2×Q8C4○D4C1
# reps116688113141

Matrix representation of C4.C25 ►in GL8(𝔽17)

01000000
160000000
00010000
001600000
000001600
00001000
000000016
00000010
,
00100000
000160000
10000000
016000000
00000001
00000010
00000100
00001000
,
00001000
00000100
00000010
00000001
160000000
016000000
001600000
000160000
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
04000000
130000000
000130000
00400000
00000400
000013000
000000013
00000040
,
04000000
130000000
00040000
001300000
00000400
000013000
00000004
000000130

G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0],[0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0] >;
 

C4.C25 in GAP, Magma, Sage, TeX 

C_4.C_2^5
 
% in TeX
 
G:=Group("C4.C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2318);
 
// by ID
 
G=gap.SmallGroup(128,2318);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,456,521,2804,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^2=f^2=1,c^2=a^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,f*b*f=a^2*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e=a^2*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁