Copied to
clipboard

## G = C2×C22.29C24order 128 = 27

### Direct product of C2 and C22.29C24

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C22.29C24
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×C22.29C24
 Lower central C1 — C22 — C2×C22.29C24
 Upper central C1 — C23 — C2×C22.29C24
 Jennings C1 — C22 — C2×C22.29C24

Generators and relations for C2×C22.29C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 1804 in 948 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×16], C4 [×8], C4 [×12], C22, C22 [×10], C22 [×104], C2×C4 [×40], C2×C4 [×28], D4 [×88], Q8 [×8], C23, C23 [×18], C23 [×104], C42 [×8], C22⋊C4 [×40], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×26], C22×C4 [×8], C2×D4 [×60], C2×D4 [×76], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×14], C24 [×16], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C42⋊C2 [×8], C22≀C2 [×32], C4⋊D4 [×32], C4.4D4 [×16], C41D4 [×16], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×22], C22×D4 [×8], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C25 [×2], C2×C42⋊C2, C2×C22≀C2 [×4], C2×C4⋊D4 [×4], C2×C4.4D4 [×2], C2×C41D4 [×2], C22.29C24 [×16], D4×C23, C22×C4○D4, C2×C22.29C24
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2+ 1+4 [×4], C25, C22.29C24 [×4], D4×C23, C2×2+ 1+4 [×2], C2×C22.29C24

Smallest permutation representation of C2×C22.29C24
On 32 points
Generators in S32
(1 19)(2 20)(3 17)(4 18)(5 15)(6 16)(7 13)(8 14)(9 22)(10 23)(11 24)(12 21)(25 32)(26 29)(27 30)(28 31)
(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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(21 29)(22 30)(23 31)(24 32)
(1 26)(2 25)(3 28)(4 27)(5 23)(6 22)(7 21)(8 24)(9 16)(10 15)(11 14)(12 13)(17 31)(18 30)(19 29)(20 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 5)(3 8)(4 7)(9 21)(10 24)(11 23)(12 22)(13 18)(14 17)(15 20)(16 19)(25 31)(26 30)(27 29)(28 32)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 25)(10 26)(11 27)(12 28)(21 31)(22 32)(23 29)(24 30)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2W 4A ··· 4H 4I ··· 4T order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 2+ 1+4 kernel C2×C22.29C24 C2×C42⋊C2 C2×C22≀C2 C2×C4⋊D4 C2×C4.4D4 C2×C4⋊1D4 C22.29C24 D4×C23 C22×C4○D4 C22×C4 C22 # reps 1 1 4 4 2 2 16 1 1 8 4

Matrix representation of C2×C22.29C24 in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [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,-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,-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,-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],[1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,2,0,-1,0,0,0,0,0,0,-1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,1,-1,0,0,0,0,2,1,-1,0,0,0,0,0,0,0,1,-2,0,0,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,0,0,-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,-1,0,1,0,0,0,0,0,2,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-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,-1,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C2×C22.29C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{29}C_2^4
% in TeX

G:=Group("C2xC2^2.29C2^4");
// GroupNames label

G:=SmallGroup(128,2178);
// by ID

G=gap.SmallGroup(128,2178);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,387,1123]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=g^2=1,e^2=b,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^-1=g*d*g=b*d=d*b,f*e*f=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,e*g=g*e,f*g=g*f>;
// generators/relations

׿
×
𝔽