Copied to
clipboard

G = C2×C22.32C24order 128 = 27

Direct product of C2 and C22.32C24

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

Aliases: C2×C22.32C24, C4211C23, C25.74C22, C22.39C25, C24.481C23, C23.272C24, C22.1022+ 1+4, C4⋊C45C23, (C2×D4)⋊7C23, (C2×Q8)⋊5C23, (C4×D4)⋊93C22, C22⋊C46C23, (C2×C4).42C24, C4⋊D464C22, (C2×C42)⋊46C22, (C23×C4)⋊32C22, (C22×C4)⋊15C23, C22⋊Q875C22, C22≀C228C22, C4.4D465C22, (C22×D4)⋊31C22, (C22×Q8)⋊26C22, C422C222C22, C2.7(C2×2+ 1+4), C23.259(C4○D4), C22.D433C22, (C2×C4×D4)⋊72C2, (C2×C4⋊D4)⋊56C2, (C2×C4⋊C4)⋊64C22, (C2×C22⋊Q8)⋊63C2, (C2×C22≀C2)⋊22C2, C22.8(C2×C4○D4), (C2×C4.4D4)⋊47C2, C2.16(C22×C4○D4), (C2×C422C2)⋊31C2, (C22×C22⋊C4)⋊31C2, (C2×C22⋊C4)⋊85C22, (C2×C22.D4)⋊50C2, SmallGroup(128,2182)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C22.32C24
C1C2C22C23C24C25C22×C22⋊C4 — C2×C22.32C24
C1C22 — C2×C22.32C24
C1C23 — C2×C22.32C24
C1C22 — C2×C22.32C24

Generators and relations for C2×C22.32C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=c, 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: 1196 in 668 conjugacy classes, 396 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×12], C4 [×20], C22, C22 [×10], C22 [×68], C2×C4 [×20], C2×C4 [×36], D4 [×36], Q8 [×4], C23, C23 [×14], C23 [×60], C42 [×8], C22⋊C4 [×56], C4⋊C4 [×24], C22×C4 [×2], C22×C4 [×24], C22×C4 [×8], C2×D4 [×28], C2×D4 [×18], C2×Q8 [×4], C2×Q8 [×2], C24, C24 [×8], C24 [×8], C2×C42 [×2], C2×C22⋊C4 [×22], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C4×D4 [×16], C22≀C2 [×16], C4⋊D4 [×24], C22⋊Q8 [×8], C22.D4 [×16], C4.4D4 [×16], C422C2 [×16], C23×C4 [×2], C23×C4 [×2], C22×D4, C22×D4 [×6], C22×Q8, C25, C22×C22⋊C4, C2×C4×D4 [×2], C2×C22≀C2 [×2], C2×C4⋊D4, C2×C4⋊D4 [×2], C2×C22⋊Q8, C2×C22.D4 [×2], C2×C4.4D4 [×2], C2×C422C2 [×2], C22.32C24 [×16], C2×C22.32C24
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ 1+4 [×4], C25, C22.32C24 [×4], C22×C4○D4, C2×2+ 1+4 [×2], C2×C22.32C24

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

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A···4H4I···4X
order12···222222···24···44···4
size11···122224···42···24···4

44 irreducible representations

dim111111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC2×C22.32C24C22×C22⋊C4C2×C4×D4C2×C22≀C2C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C422C2C22.32C24C23C22
# reps1122312221684

Matrix representation of C2×C22.32C24 in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
10000000
00010000
00100000
00000010
00001414
00001000
00000001
,
20000000
02000000
00200000
00020000
00000100
00001000
00004141
00003201
,
10000000
04000000
00400000
00010000
00001000
00000400
00000010
00002024
,
40000000
04000000
00100000
00010000
00001000
00000100
00000040
00002304

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

C2×C22.32C24 in GAP, Magma, Sage, TeX

C_2\times C_2^2._{32}C_2^4
% in TeX

G:=Group("C2xC2^2.32C2^4");
// GroupNames label

G:=SmallGroup(128,2182);
// by ID

G=gap.SmallGroup(128,2182);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,387,1123]);
// 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=c,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