Copied to
clipboard

G = C23.257C24  order 128 = 27

110th central extension by C23 of C24

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

Aliases: C25.25C22, C23.257C24, C24.224C23, C22.882+ (1+4), C22≀C2⋊9C4, C24⋊13(C2×C4), C24⋊3C4⋊8C2, (C23×C4)⋊8C22, (C2×C42)⋊17C22, C23.26(C22×C4), C23.23D4⋊17C2, C22.148(C23×C4), (C22×C4).485C23, C24.3C22⋊23C2, C24.C22⋊25C2, C2.C42⋊13C22, C2.1(C24⋊C22), (C22×D4).112C22, C2.37(C22.11C24), C2.1(C22.54C24), (C2×C4⋊C4)⋊9C22, C22⋊C4⋊19(C2×C4), (C2×C22≀C2).6C2, (C2×D4).133(C2×C4), (C2×C22⋊C4)⋊5C22, (C2×C4).54(C22×C4), SmallGroup(128,1107)

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

Derived series
C1 — C22 — C23.257C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C23.257C24
Lower central
C1 — C22 — C23.257C24
Upper central
C1 — C23 — C23.257C24
Jennings
C1 — C23 — C23.257C24

Subgroups: 812 in 344 conjugacy classes, 132 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×14], C22, C22 [×6], C22 [×56], C2×C4 [×6], C2×C4 [×34], D4 [×12], C23, C23 [×8], C23 [×52], C42 [×3], C22⋊C4 [×12], C22⋊C4 [×15], C4⋊C4 [×3], C22×C4 [×11], C22×C4 [×3], C2×D4 [×12], C2×D4 [×6], C24, C24 [×7], C24 [×6], C2.C42 [×6], C2×C42 [×3], C2×C22⋊C4 [×18], C2×C4⋊C4 [×3], C22≀C2 [×8], C23×C4, C22×D4 [×3], C25, C24⋊3C4 [×2], C23.23D4 [×3], C24.C22 [×6], C24.3C22 [×3], C2×C22≀C2, C23.257C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ (1+4) [×6], C22.11C24 [×3], C22.54C24 [×3], C24⋊C22, C23.257C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ede=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, gdg=abd, fef=abe, fg=gf >

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

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

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

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

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

Matrix representation ►G ⊆ GL12(ℤ)

-100000000000
0-10000000000
00-1000000000
000-100000000
0000-10000000
00000-1000000
000000-100000
0000000-10000
00000000-1000
000000000-100
0000000000-10
00000000000-1
,
-100000000000
0-10000000000
00-1000000000
000-100000000
0000-10000000
00000-1000000
000000-100000
0000000-10000
000000001000
000000000100
000000000010
000000000001
,
-100000000000
0-10000000000
00-1000000000
000-100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
01-2000000000
100-200000000
100-100000000
01-1000000000
000012-200000
00000-11-10000
0000000-10000
000000-100000
0000000001-10
00000000100-1
00000000000-1
0000000000-10
,
010000000000
100000000000
100-100000000
01-1000000000
000012000000
00000-1000000
0000100-10000
000012-100000
000000000100
000000001000
00000000200-1
0000000002-10
,
100000000000
010000000000
01-1000000000
100-100000000
0000-10000000
00000-1000000
0000-1-2100000
0000-10010000
000000001000
000000000-100
000000000-210
00000000200-1
,
-100000000000
010000000000
01-1000000000
-100100000000
000010000000
0000-1-1000000
0000-1-2100000
0000100-10000
000000001000
000000000-100
0000000000-10
000000000001

G:=sub<GL(12,Integers())| [-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,-2,0,0,-1,0,0,0,0,0,0,0,0,0,-2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,0,-2,1,0,-1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,0,0,-1,-1,0],[0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,2,-1,0,2,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0],[1,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,-1,0,0,0,0,0,0,0,0,0,-1,-2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,-1,-2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,-1,1,0,0,0,0,0,0,0,0,0,-1,-2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1] >;
 

38 conjugacy classes

class 1 2A···2G2H···2O4A···4V
order12···22···24···4
size11···14···44···4

38 irreducible representations

dim11111114
type+++++++
imageC1C2C2C2C2C2C42+ (1+4)
kernelC23.257C24C24⋊3C4C23.23D4C24.C22C24.3C22C2×C22≀C2C22≀C2C22
# reps123631166

In GAP, Magma, Sage, TeX 

C_2^3._{257}C_2^4
 
% in TeX
 
G:=Group("C2^3.257C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1107);
 
// by ID
 
G=gap.SmallGroup(128,1107);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,1571,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=c,a*b=b*a,a*c=c*a,e*d*e=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=a*b*d,f*e*f=a*b*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀