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G = C22.128C25  order 128 = 27

109th central stem extension by C22 of C25

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

Aliases: C23.69C24, C42.111C23, C24.517C23, C22.128C25, C4.872+ (1+4), C4⋊Q8⋊41C22, D4⋊6D4⋊34C2, Q8⋊6D4⋊25C2, (C4×D4)⋊62C22, (C4×Q8)⋊59C22, C4⋊D4⋊36C22, C4⋊C4.316C23, C4⋊1D4⋊24C22, (C2×C4).118C24, C22⋊Q8⋊46C22, (C2×D4).320C23, C22⋊C4.45C23, (C2×Q8).462C23, C42.C2⋊22C22, C22.19C24⋊41C2, C42⋊C2⋊56C22, C22.54C24⋊6C2, C42⋊2C2⋊15C22, C22≀C2.32C22, (C23×C4).619C22, (C22×C4).388C23, C2.57(C2×2+ (1+4)), C2.46(C2.C25), C22.D4⋊18C22, C22.33C24⋊15C2, C22.46C24⋊30C2, C22.31C24⋊21C2, C22.35C24⋊17C2, C22.47C24⋊29C2, C22.34C24⋊19C2, (C2×C4⋊C4)⋊87C22, (C2×C4○D4)⋊46C22, SmallGroup(128,2271)

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

Derived series
C1 — C22 — C22.128C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.128C25
Lower central
C1 — C22 — C22.128C25
Upper central
C1 — C22 — C22.128C25
Jennings
C1 — C22 — C22.128C25

Subgroups: 828 in 525 conjugacy classes, 380 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×9], C4 [×2], C4 [×21], C22, C22 [×31], C2×C4 [×2], C2×C4 [×20], C2×C4 [×29], D4 [×35], Q8 [×5], C23, C23 [×8], C23 [×3], C42 [×2], C42 [×8], C22⋊C4 [×42], C4⋊C4 [×42], C22×C4 [×2], C22×C4 [×22], C22×C4 [×2], C2×D4, C2×D4 [×32], C2×Q8, C2×Q8 [×2], C4○D4 [×10], C24, C2×C4⋊C4 [×6], C42⋊C2, C42⋊C2 [×6], C4×D4 [×22], C4×Q8 [×2], C22≀C2 [×6], C4⋊D4 [×40], C22⋊Q8 [×12], C22.D4 [×26], C42.C2 [×2], C42.C2 [×8], C42⋊2C2 [×12], C4⋊1D4, C4⋊1D4 [×4], C4⋊Q8, C23×C4, C2×C4○D4, C2×C4○D4 [×4], C22.19C24, C22.19C24 [×2], C22.31C24 [×2], C22.33C24 [×4], C22.34C24, C22.34C24 [×4], C22.35C24, D4⋊6D4 [×2], Q8⋊6D4 [×2], C22.46C24 [×2], C22.47C24 [×6], C22.54C24 [×4], C22.128C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×2], C25, C2×2+ (1+4), C2.C25 [×2], C22.128C25

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

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

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

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

(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)

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
01000000
10000000
00040000
00400000
00000100
00001000
00000001
00000010
,
30000000
03000000
00300000
00030000
00004000
00000100
00000010
00000004
,
10000000
04000000
00100000
00040000
00001000
00000100
00000040
00000004
,
20000000
02000000
00300000
00030000
00001000
00000100
00000010
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[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],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

38 conjugacy classes

class 1 2A2B2C2D···2L4A···4F4G···4Y
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim1111111111144
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C22+ (1+4)C2.C25
kernelC22.128C25C22.19C24C22.31C24C22.33C24C22.34C24C22.35C24D4⋊6D4Q8⋊6D4C22.46C24C22.47C24C22.54C24C4C2
# reps1324512226424

In GAP, Magma, Sage, TeX 

C_2^2._{128}C_2^5
 
% in TeX
 
G:=Group("C2^2.128C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2271);
 
// by ID
 
G=gap.SmallGroup(128,2271);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,184,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=g^2=a,a*b=b*a,d*c*d=g*c*g^-1=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=f*c*f=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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