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G = C22.49C25  order 128 = 27

30th central stem extension by C22 of C25

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

Aliases: C22.49C25, C42.550C23, C24.489C23, C23.121C24, C4.1852+ 1+4, C4⋊Q8⋊80C22, (C4×D4)⋊99C22, (C2×C4).51C24, (C4×Q8)⋊89C22, C4⋊D4⋊67C22, C4⋊C4.463C23, (C2×C42)⋊49C22, C22⋊Q8⋊81C22, (C2×D4).295C23, C4.4D4⋊69C22, C22⋊C4.79C23, (C2×Q8).427C23, C42.C2⋊46C22, C4○2(C22.29C24), C4○3(C22.32C24), C22.32C24⋊30C2, C22.19C24⋊16C2, C42⋊C2⋊92C22, C22.29C24⋊38C2, C42⋊2C2⋊27C22, C22≀C2.22C22, C4⋊1D4.180C22, C2.8(C2.C25), (C23×C4).592C22, C2.14(C2×2+ 1+4), C22.26C24⋊28C2, (C22×C4).1188C23, (C22×D4).588C22, C22.D4⋊39C22, C4○2(C22.36C24), C4○2(C22.31C24), C4○2(C23.41C23), C4○2(C22.34C24), C23.41C23⋊29C2, C22.31C24⋊31C2, C22.36C24⋊46C2, C23.36C23⋊19C2, C22.34C24⋊31C2, (C2×C4×D4)⋊77C2, (C4×C4○D4)⋊18C2, (C2×C4)⋊5(C4○D4), C4.76(C2×C4○D4), (C2×C4○D4)⋊20C22, C22.12(C2×C4○D4), C2.23(C22×C4○D4), (C2×C4⋊C4).952C22, (C2×C22⋊C4).537C22, (C2×C4)○(C22.34C24), (C2×C4)○(C23.41C23), SmallGroup(128,2192)

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

Derived series
C1 — C22 — C22.49C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C22.49C25
Lower central
C1 — C22 — C22.49C25
Upper central
C1 — C2×C4 — C22.49C25
Jennings
C1 — C22 — C22.49C25

Generators and relations for C22.49C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=a, g2=b, ab=ba, dcd-1=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: 884 in 570 conjugacy classes, 390 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C4×C4○D4, C22.19C24, C23.36C23, C22.26C24, C22.29C24, C22.31C24, C22.32C24, C22.34C24, C22.36C24, C23.41C23, C22.49C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C25, C22×C4○D4, C2×2+ 1+4, C2.C25, C22.49C25

Smallest permutation representation of C22.49C25
►On 32 points
Generators in S32
(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 13)(2 14)(3 15)(4 16)(5 20)(6 17)(7 18)(8 19)(9 27)(10 28)(11 25)(12 26)(21 29)(22 30)(23 31)(24 32)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A4B4C4D4E···4N4O···4AD
order1222222···244444···44···4
size1111224···411112···24···4

44 irreducible representations

dim111111111111244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4○D42+ 1+4C2.C25
kernelC22.49C25C2×C4×D4C4×C4○D4C22.19C24C23.36C23C22.26C24C22.29C24C22.31C24C22.32C24C22.34C24C22.36C24C23.41C23C2×C4C4C2
# reps112444214441822

Matrix representation of C22.49C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
040000
400000
000010
000001
001000
000100
,
400000
040000
001100
003400
000044
000021
,
400000
010000
001000
003400
000040
000021
,
100000
010000
004000
000400
000010
000001
,
300000
030000
002000
000200
000020
000002

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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,3,0,0,0,0,1,4,0,0,0,0,0,0,4,2,0,0,0,0,4,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,4,0,0,0,0,0,0,4,2,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
 

C22.49C25 in GAP, Magma, Sage, TeX 

C_2^2._{49}C_2^5
 
% in TeX
 
G:=Group("C2^2.49C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2192);
 
// by ID
 
G=gap.SmallGroup(128,2192);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,387,1123,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=a,g^2=b,a*b=b*a,d*c*d^-1=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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