Copied to
clipboard

## G = C2×C22.54C24order 128 = 27

### Direct product of C2 and C22.54C24

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

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

Generators and relations for C2×C22.54C24
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 >

Subgroups: 1308 in 672 conjugacy classes, 388 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22≀C2, C4⋊D4, C22.D4, C422C2, C41D4, C23×C4, C22×D4, C25, C2×C22≀C2, C2×C4⋊D4, C2×C22.D4, C2×C422C2, C2×C41D4, C22.54C24, C2×C22.54C24
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C22.54C24, C2×2+ 1+4, C2×C22.54C24

Smallest permutation representation of C2×C22.54C24
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 19)(2 20)(3 5)(4 6)(7 10)(8 9)(11 18)(12 17)(13 31)(14 32)(15 29)(16 30)(21 28)(22 27)(23 26)(24 25)
(1 4)(2 3)(5 20)(6 19)(7 12)(8 11)(9 18)(10 17)(13 22)(14 21)(15 23)(16 24)(25 30)(26 29)(27 31)(28 32)
(1 10)(2 9)(3 18)(4 17)(5 11)(6 12)(7 19)(8 20)(13 30)(14 29)(15 32)(16 31)(21 26)(22 25)(23 28)(24 27)
(1 24)(2 23)(3 15)(4 16)(5 29)(6 30)(7 27)(8 28)(9 21)(10 22)(11 32)(12 31)(13 17)(14 18)(19 25)(20 26)
(1 4)(2 3)(5 20)(6 19)(13 27)(14 28)(15 29)(16 30)(21 32)(22 31)(23 26)(24 25)
(7 17)(8 18)(9 11)(10 12)(13 22)(14 21)(15 29)(16 30)(23 26)(24 25)(27 31)(28 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), (1,19)(2,20)(3,5)(4,6)(7,10)(8,9)(11,18)(12,17)(13,31)(14,32)(15,29)(16,30)(21,28)(22,27)(23,26)(24,25), (1,4)(2,3)(5,20)(6,19)(7,12)(8,11)(9,18)(10,17)(13,22)(14,21)(15,23)(16,24)(25,30)(26,29)(27,31)(28,32), (1,10)(2,9)(3,18)(4,17)(5,11)(6,12)(7,19)(8,20)(13,30)(14,29)(15,32)(16,31)(21,26)(22,25)(23,28)(24,27), (1,24)(2,23)(3,15)(4,16)(5,29)(6,30)(7,27)(8,28)(9,21)(10,22)(11,32)(12,31)(13,17)(14,18)(19,25)(20,26), (1,4)(2,3)(5,20)(6,19)(13,27)(14,28)(15,29)(16,30)(21,32)(22,31)(23,26)(24,25), (7,17)(8,18)(9,11)(10,12)(13,22)(14,21)(15,29)(16,30)(23,26)(24,25)(27,31)(28,32)>;

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4R order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 2+ 1+4 kernel C2×C22.54C24 C2×C22≀C2 C2×C4⋊D4 C2×C22.D4 C2×C42⋊2C2 C2×C4⋊1D4 C22.54C24 C22 # reps 1 3 6 3 2 1 16 6

Matrix representation of C2×C22.54C24 in GL12(ℤ)

 -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 2 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 -1 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 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 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 -1 0 1 0 0 0 0 0 0 0 0 1 -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 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 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 0 0 0 0 0 0 0 1 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 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 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 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

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] >;

C2×C22.54C24 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

׿
×
𝔽