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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, Q8oD8:7C2, Q8o(C8:C22), D4oSD16:6C2, C4oD4.55D4, D4.66(C2xD4), C4oD8:4C22, C8oD4:8C22, Q8.68(C2xD4), D4o(C8.C22), (C2xD4).338D4, Q8oM4(2):4C2, (C2xQ8).257D4, C2.48(D4xC23), C8:C22:16C22, (C2xC4).150C24, (C2xC8).125C23, C2.C25:7C2, (C2xQ16):35C22, C4oD4.18C23, C4.130(C22xD4), C23.359(C2xD4), D8:C22:12C2, (C2xSD16):37C22, (C2xD4).348C23, C8.C22:16C22, (C2xQ8).325C23, (C22xQ8):49C22, C22.22(C22xD4), (C2xM4(2)):35C22, (C22xC4).418C23, (C2x2- 1+4):13C2, (C2xC4).671(C2xD4), (C2xC4oD4):60C22, (C2xC8.C22):36C2, SmallGroup(128,2318)

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

Derived series
C1 — C4 — C4.C25
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC4oD4 — C2x2- 1+4 — C4.C25
Lower central
C1 — C2 — C4 — C4.C25
Upper central
C1 — C2 — C2xC4oD4 — 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, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C2xM4(2), C8oD4, C2xSD16, C2xQ16, C4oD8, C8:C22, C8.C22, C22xQ8, C22xQ8, C2xC4oD4, C2xC4oD4, C2xC4oD4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, Q8oM4(2), C2xC8.C22, D8:C22, D4oSD16, Q8oD8, C2x2- 1+4, C2.C25, C4.C25
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22xD4, C25, D4xC23, 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.C25Q8oM4(2)C2xC8.C22D8:C22D4oSD16Q8oD8C2x2- 1+4C2.C25C2xD4C2xQ8C4oD4C1
# 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
 

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