Copied to
clipboard

?

G = C22.125C25  order 128 = 27

106th central stem extension by C22 of C25

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

Aliases: C23.66C24, C22.125C25, C42.108C23, C24.145C23, C4.842+ (1+4), C22.102- (1+4), C22.212+ (1+4), C4⋊Q8⋊39C22, D4⋊5D4⋊31C2, D4⋊6D4⋊33C2, D4⋊3Q8⋊33C2, Q8⋊5D4⋊26C2, (C4×D4)⋊60C22, (C4×Q8)⋊57C22, C23⋊2Q8⋊8C2, C4⋊D4⋊35C22, C4⋊C4.313C23, (C2×C4).115C24, C22⋊Q8⋊44C22, (C2×D4).317C23, C4.4D4⋊36C22, C22⋊C4.43C23, (C2×Q8).461C23, C42.C2⋊19C22, (C22×Q8)⋊39C22, C42⋊C2⋊53C22, C22.19C24⋊39C2, C22.32C24⋊14C2, C42⋊2C2⋊12C22, C22≀C2.13C22, (C23×C4).617C22, (C22×C4).385C23, C2.40(C2×2- (1+4)), C2.54(C2×2+ (1+4)), C22.56C24⋊5C2, (C22×D4).437C22, C22.D4⋊15C22, C22.36C24⋊26C2, C22.33C24⋊13C2, C22.31C24⋊20C2, (C2×C4⋊D4)⋊73C2, (C2×C4⋊C4)⋊85C22, (C2×C22⋊Q8)⋊83C2, (C2×C4○D4)⋊44C22, (C2×C22⋊C4).390C22, SmallGroup(128,2268)

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

Derived series
C1 — C22 — C22.125C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.125C25
Lower central
C1 — C22 — C22.125C25
Upper central
C1 — C22 — C22.125C25
Jennings
C1 — C22 — C22.125C25

Subgroups: 940 in 567 conjugacy classes, 384 normal (38 characteristic)
C1, C2 [×3], C2 [×11], C4 [×2], C4 [×21], C22, C22 [×4], C22 [×33], C2×C4 [×4], C2×C4 [×18], C2×C4 [×33], D4 [×37], Q8 [×11], C23 [×3], C23 [×6], C23 [×13], C42 [×6], C22⋊C4 [×46], C4⋊C4 [×2], C4⋊C4 [×36], C22×C4 [×6], C22×C4 [×20], C22×C4 [×2], C2×D4, C2×D4 [×26], C2×D4 [×6], C2×Q8, C2×Q8 [×8], C2×Q8 [×2], C4○D4 [×14], C24, C24 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2, C4×D4 [×14], C4×Q8 [×2], C22≀C2 [×6], C4⋊D4 [×2], C4⋊D4 [×30], C22⋊Q8 [×2], C22⋊Q8 [×34], C22.D4 [×18], C4.4D4 [×10], C42.C2 [×4], C42⋊2C2 [×4], C4⋊Q8 [×2], C23×C4, C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4, C2×C4○D4 [×6], C2×C4⋊D4, C2×C22⋊Q8, C22.19C24, C22.31C24 [×4], C22.32C24 [×2], C22.33C24 [×2], C22.36C24 [×2], C23⋊2Q8 [×2], D4⋊5D4 [×6], D4⋊6D4 [×2], Q8⋊5D4 [×2], D4⋊3Q8 [×2], C22.56C24 [×4], C22.125C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×4], 2- (1+4) [×2], C25, C2×2+ (1+4) [×2], C2×2- (1+4), C22.125C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, ab=ba, dcd=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=fcf=bc=cb, ede=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 11)(2 12)(3 9)(4 10)(5 29)(6 30)(7 31)(8 32)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
01000000
10000000
10040000
01400000
00000100
00001000
00000004
00000040
,
01300000
10030000
00040000
00400000
00000030
00000003
00002000
00000200
,
40000000
04000000
00400000
00040000
00000010
00000001
00001000
00000100
,
40000000
04000000
04100000
40010000
00000010
00000001
00001000
00000100
,
01000000
40000000
40010000
01400000
00000100
00004000
00000001
00000040

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2N4A4B4C···4W
order122222222···2444···4
size111122224···4224···4

38 irreducible representations

dim11111111111111444
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C22+ (1+4)2+ (1+4)2- (1+4)
kernelC22.125C25C2×C4⋊D4C2×C22⋊Q8C22.19C24C22.31C24C22.32C24C22.33C24C22.36C24C23⋊2Q8D4⋊5D4D4⋊6D4Q8⋊5D4D4⋊3Q8C22.56C24C4C22C22
# reps11114222262224222

In GAP, Magma, Sage, TeX 

C_2^2._{125}C_2^5
 
% in TeX
 
G:=Group("C2^2.125C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2268);
 
// by ID
 
G=gap.SmallGroup(128,2268);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,2019,570,1684,102]);
 
// 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=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=f*c*f=b*c=c*b,e*d*e=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
 

׿
×
⋊
ℤ
𝔽
○
≀