Copied to
clipboard

G = C2×C232Q8order 128 = 27

Direct product of C2 and C232Q8

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

Aliases: C2×C232Q8, C247Q8, C25.75C22, C22.45C25, C24.613C23, C23.117C24, C22.1062+ 1+4, C4⋊C46C23, C234(C2×Q8), (C2×Q8)⋊6C23, C2.7(Q8×C23), (C2×C4).47C24, C22⋊Q879C22, C22.5(C22×Q8), C22⋊C4.75C23, (C22×Q8)⋊27C22, (C23×C4).589C22, C2.11(C2×2+ 1+4), (C22×C4).1185C23, (C2×C4⋊C4)⋊65C22, (C2×C22⋊Q8)⋊66C2, (C22×C22⋊C4).29C2, (C2×C22⋊C4).534C22, SmallGroup(128,2188)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C232Q8
C1C2C22C23C24C23×C4C22×C22⋊C4 — C2×C232Q8
C1C22 — C2×C232Q8
C1C23 — C2×C232Q8
C1C22 — C2×C232Q8

Generators and relations for C2×C232Q8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 1068 in 648 conjugacy classes, 436 normal (6 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×24], C22, C22 [×18], C22 [×52], C2×C4 [×24], C2×C4 [×48], Q8 [×16], C23, C23 [×34], C23 [×36], C22⋊C4 [×48], C4⋊C4 [×48], C22×C4 [×36], C22×C4 [×12], C2×Q8 [×16], C2×Q8 [×8], C24 [×15], C24 [×4], C2×C22⋊C4 [×36], C2×C4⋊C4 [×12], C22⋊Q8 [×96], C23×C4 [×6], C22×Q8 [×4], C25, C22×C22⋊C4 [×3], C2×C22⋊Q8 [×12], C232Q8 [×16], C2×C232Q8
Quotients: C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], C2×Q8 [×28], C24 [×31], C22×Q8 [×14], 2+ 1+4 [×4], C25, C232Q8 [×4], Q8×C23, C2×2+ 1+4 [×2], C2×C232Q8

Smallest permutation representation of C2×C232Q8
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 23)(6 24)(7 21)(8 22)(13 17)(14 18)(15 19)(16 20)(25 32)(26 29)(27 30)(28 31)
(1 3)(2 21)(4 23)(5 12)(6 8)(7 10)(9 11)(13 28)(14 16)(15 26)(17 31)(18 20)(19 29)(22 24)(25 27)(30 32)
(13 26)(14 27)(15 28)(16 25)(17 29)(18 30)(19 31)(20 32)
(1 22)(2 23)(3 24)(4 21)(5 10)(6 11)(7 12)(8 9)(13 26)(14 27)(15 28)(16 25)(17 29)(18 30)(19 31)(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 19 3 17)(2 18 4 20)(5 27 7 25)(6 26 8 28)(9 15 11 13)(10 14 12 16)(21 32 23 30)(22 31 24 29)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2S4A···4X
order12···22···24···4
size11···12···24···4

44 irreducible representations

dim111124
type++++-+
imageC1C2C2C2Q82+ 1+4
kernelC2×C232Q8C22×C22⋊C4C2×C22⋊Q8C232Q8C24C22
# reps13121684

Matrix representation of C2×C232Q8 in GL8(𝔽5)

10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000100
00000010
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
04000000
10000000
00010000
00400000
00000400
00001000
00000001
00000040
,
20000000
03000000
00200000
00030000
00000040
00000004
00001000
00000100

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

C2×C232Q8 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_2Q_8
% in TeX

G:=Group("C2xC2^3:2Q8");
// GroupNames label

G:=SmallGroup(128,2188);
// by ID

G=gap.SmallGroup(128,2188);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,387,352,1123]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽