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G = C22.48C25  order 128 = 27

29th central stem extension by C22 of C25

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

Aliases: C22.48C25, C24.488C23, C23.120C24, C42.549C23, C4.1842+ 1+4, (C4xD4):98C22, C23:3(C4oD4), (C2xC4).50C24, (C4xQ8):33C22, C4o2(C23:3D4), C4:D4:66C22, C23:3D4:19C2, C4:C4.288C23, (C23xC4):33C22, (C2xC42):48C22, C4o2(C23:2Q8), C23:2Q8:11C2, C22:Q8:80C22, C22wrC2:29C22, (C2xD4).294C23, C4.4D4:68C22, C22:C4.78C23, (C2xQ8).278C23, C42.C2:45C22, C4o2(C22.32C24), C22.19C24:15C2, C42:2C2:26C22, C42:C2:28C22, C22.32C24:29C2, C2.7(C2.C25), C2.13(C2x2+ 1+4), (C22xC4).1187C23, (C22xD4).587C22, C22.D4:38C22, C4o2(C22.33C24), C23.36C23:18C2, C22.33C24:29C2, (C2xC4xD4):76C2, (C2xC4:C4):131C22, (C2xC4oD4):19C22, (C2xC4)o(C23:2Q8), C22.11(C2xC4oD4), C2.22(C22xC4oD4), (C2xC22:C4):86C22, (C2xC4)o(C22.33C24), SmallGroup(128,2191)

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

Derived series
C1 — C22 — C22.48C25
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C23xC4 — C2xC4xD4 — C22.48C25
Lower central
C1 — C22 — C22.48C25
Upper central
C1 — C2xC4 — C22.48C25
Jennings
C1 — C22 — C22.48C25

Generators and relations for C22.48C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=b, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 916 in 580 conjugacy classes, 390 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C24, C2xC42, C2xC22:C4, C2xC4:C4, C42:C2, C4xD4, C4xQ8, C22wrC2, C4:D4, C22:Q8, C22.D4, C4.4D4, C42.C2, C42:2C2, C23xC4, C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, C22.19C24, C23.36C23, C23:3D4, C22.32C24, C22.33C24, C23:2Q8, C22.48C25
Quotients: C1, C2, C22, C23, C4oD4, C24, C2xC4oD4, 2+ 1+4, C25, C22xC4oD4, C2x2+ 1+4, C2.C25, C22.48C25

Smallest permutation representation of C22.48C25
►On 32 points
Generators in S32
(1 11)(2 12)(3 9)(4 10)(5 32)(6 29)(7 30)(8 31)(13 20)(14 17)(15 18)(16 19)(21 28)(22 25)(23 26)(24 27)

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

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

(5 30)(6 31)(7 32)(8 29)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)(25 27)(26 28)

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2O4A4B4C4D4E···4J4K···4AB
order12222···22···244444···44···4
size11112···24···411112···24···4

44 irreducible representations

dim11111111244
type+++++++++
imageC1C2C2C2C2C2C2C2C4oD42+ 1+4C2.C25
kernelC22.48C25C2xC4xD4C22.19C24C23.36C23C23:3D4C22.32C24C22.33C24C23:2Q8C23C4C2
# reps13663661822

Matrix representation of C22.48C25 ►in GL6(F5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
001000
000001
003040
000100
,
100000
010000
004040
003140
000010
000044
,
100000
040000
001411
000400
000043
000001
,
100000
010000
004040
000400
000010
000001
,
300000
030000
003000
000300
000030
000003

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

C22.48C25 in GAP, Magma, Sage, TeX 

C_2^2._{48}C_2^5
 
% in TeX
 
G:=Group("C2^2.48C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2191);
 
// by ID
 
G=gap.SmallGroup(128,2191);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,387,1123,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=b,a*b=b*a,d*c*d=f*c*f=a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*g=g*c,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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