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G = C22.74C25  order 128 = 27

55th central stem extension by C22 of C25

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

Aliases: C23.34C24, C22.74C25, C42.76C23, C24.501C23, (C2×D4)⋊54D4, D4.57(C2×D4), C4⋊Q8⋊28C22, D4⋊6D4⋊17C2, D4⋊5D4⋊13C2, (C4×D4)⋊33C22, C23⋊3D4⋊5C2, (C2×C4).68C24, C2.26(D4×C23), C22≀C2⋊4C22, C4⋊D4⋊79C22, C4⋊C4.290C23, (C23×C4)⋊39C22, D4○(C22.D4), C23.356(C2×D4), C4.115(C22×D4), C22⋊Q8⋊22C22, (C2×D4).463C23, C4.4D4⋊20C22, (C22×D4)⋊33C22, C22⋊C4.15C23, (C2×2+ 1+4)⋊8C2, (C2×Q8).439C23, (C22×Q8)⋊65C22, C22.10(C22×D4), C42⋊C2⋊31C22, C22.19C24⋊21C2, C22.11C24⋊14C2, (C22×C4).350C23, C2.14(C2.C25), C22.D4⋊68C22, C23.38C23⋊18C2, (C2×C4⋊C4)⋊67C22, (C2×C4).663(C2×D4), (C2×C4○D4)⋊75C22, (C22×C4○D4)⋊21C2, (C2×C22⋊C4)⋊44C22, (C2×D4)○(C22.D4), (C2×C22.D4)⋊59C2, SmallGroup(128,2217)

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

Derived series
C1 — C22 — C22.74C25
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4○D4 — C22.74C25
Lower central
C1 — C22 — C22.74C25
Upper central
C1 — C22 — C22.74C25
Jennings
C1 — C22 — C22.74C25

Generators and relations for C22.74C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=a, 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: 1276 in 780 conjugacy classes, 428 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, 2+ 1+4, C22.11C24, C2×C22.D4, C22.19C24, C23⋊3D4, C23.38C23, D4⋊5D4, D4⋊6D4, C22×C4○D4, C2×2+ 1+4, C22.74C25
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D4×C23, C2.C25, C22.74C25

Smallest permutation representation of C22.74C25
►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 15)(2 16)(3 13)(4 14)(5 19)(6 20)(7 17)(8 18)(9 25)(10 26)(11 27)(12 28)(21 29)(22 30)(23 31)(24 32)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2M2N···2S4A···4H4I···4X
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C2.C25
kernelC22.74C25C22.11C24C2×C22.D4C22.19C24C23⋊3D4C23.38C23D4⋊5D4D4⋊6D4C22×C4○D4C2×2+ 1+4C2×D4C2
# reps114242881184

Matrix representation of C22.74C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
010000
001003
004041
004401
000004
,
400000
040000
004030
000011
000010
000140
,
040000
400000
003000
000300
000030
000003
,
100000
010000
004300
000100
001104
004440
,
400000
040000
004300
000100
000101
000410

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

C22.74C25 in GAP, Magma, Sage, TeX 

C_2^2._{74}C_2^5
 
% in TeX
 
G:=Group("C2^2.74C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2217);
 
// by ID
 
G=gap.SmallGroup(128,2217);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,184,570,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=g^2=1,e^2=a,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
 

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