Copied to
clipboard

?

G = C2×C22.54C24  order 128 = 27

Direct product of C2 and C22.54C24

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

Aliases: C2×C22.54C24, C42⋊15C23, C25.79C22, C23.57C24, C22.114C25, C24.512C23, C22.1172+ (1+4), C4⋊C4⋊9C23, (C2×D4)⋊9C23, C4⋊D4⋊87C22, C4⋊1D4⋊54C22, C22⋊C4⋊10C23, (C2×C4).104C24, (C23×C4)⋊46C22, (C2×C42)⋊68C22, (C22×C4)⋊18C23, C22≀C2⋊36C22, (C22×D4)⋊40C22, C42⋊2C2⋊40C22, C2.45(C2×2+ (1+4)), C22.D4⋊57C22, (C2×C4⋊D4)⋊70C2, (C2×C4⋊1D4)⋊28C2, (C2×C4⋊C4)⋊81C22, (C2×C22≀C2)⋊27C2, (C2×C42⋊2C2)⋊39C2, (C2×C22⋊C4)⋊54C22, (C2×C22.D4)⋊62C2, SmallGroup(128,2257)

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

Derived series
C1 — C22 — C2×C22.54C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C2×C22.54C24
Lower central
C1 — C22 — C2×C22.54C24
Upper central
C1 — C23 — C2×C22.54C24
Jennings
C1 — C22 — C2×C22.54C24

Subgroups: 1308 in 672 conjugacy classes, 388 normal (8 characteristic)
C1, C2 [×7], C2 [×12], C4 [×18], C22, C22 [×6], C22 [×76], C2×C4 [×18], C2×C4 [×30], D4 [×48], C23, C23 [×12], C23 [×64], C42 [×4], C22⋊C4 [×48], C4⋊C4 [×24], C22×C4 [×21], C22×C4 [×6], C2×D4 [×48], C2×D4 [×24], C24, C24 [×9], C24 [×6], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C22≀C2 [×24], C4⋊D4 [×48], C22.D4 [×24], C42⋊2C2 [×16], C4⋊1D4 [×8], C23×C4 [×3], C22×D4 [×12], C25, C2×C22≀C2 [×3], C2×C4⋊D4 [×6], C2×C22.D4 [×3], C2×C42⋊2C2 [×2], C2×C4⋊1D4, C22.54C24 [×16], C2×C22.54C24

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×6], C25, C22.54C24 [×4], C2×2+ (1+4) [×3], C2×C22.54C24

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

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

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

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

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

(1 12)(2 11)(3 5)(4 6)(7 25)(8 26)(9 14)(10 13)(15 20)(16 19)(29 32)(30 31)

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

Matrix representation ►G ⊆ GL12(ℤ)

-100000000000
0-10000000000
00-1000000000
000-100000000
000010000000
000001000000
000000100000
000000010000
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
0000-10000000
00000-1000000
000000-100000
0000000-10000
00000000-1000
000000000-100
0000000000-10
00000000000-1
,
102000000000
001-100000000
00-1000000000
0-1-1000000000
000000100000
000000010000
000010000000
000001000000
000000000010
000000000001
000000001000
000000000100
,
-120000000000
010000000000
1-10100000000
1-11000000000
000001000000
000010000000
0000000-10000
000000-100000
000000000100
000000001000
000000000001
000000000010
,
-100000000000
0-10000000000
101000000000
100100000000
0000-10000000
00000-1000000
000000100000
000000010000
00000000-1000
000000000100
000000000010
00000000000-1
,
100000000000
1-10000000000
001000000000
-100-100000000
000010000000
00000-1000000
000000100000
0000000-10000
000000001000
000000000100
0000000000-10
00000000000-1

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],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,2,1,-1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[-1,0,1,1,0,0,0,0,0,0,0,0,2,1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[-1,0,1,1,0,0,0,0,0,0,0,0,0,-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],[1,1,0,-1,0,0,0,0,0,0,0,0,0,-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] >;
 

38 conjugacy classes

class 1 2A···2G2H···2S4A···4R
order12···22···24···4
size11···14···44···4

38 irreducible representations

dim11111114
type++++++++
imageC1C2C2C2C2C2C22+ (1+4)
kernelC2×C22.54C24C2×C22≀C2C2×C4⋊D4C2×C22.D4C2×C42⋊2C2C2×C4⋊1D4C22.54C24C22
# reps136321166

In GAP, Magma, Sage, TeX 

C_2\times C_2^2._{54}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.54C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2257);
 
// by ID
 
G=gap.SmallGroup(128,2257);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,1059,2915,570]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f=b*c*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀