Copied to
clipboard

G = C2×C8.C22order 64 = 26

Direct product of C2 and C8.C22

direct product, p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: C2×C8.C22, C4.6C24, C8.1C23, Q163C22, D4.3C23, C23.51D4, Q8.3C23, SD162C22, M4(2)⋊4C22, C4.65(C2×D4), (C2×C4).50D4, (C2×Q16)⋊11C2, (C2×SD16)⋊5C2, (C22×Q8)⋊9C2, (C2×M4(2))⋊4C2, (C2×C8).25C22, (C2×C4).41C23, (C2×Q8)⋊15C22, C2.28(C22×D4), C22.24(C2×D4), C4○D4.12C22, (C2×D4).74C22, (C22×C4).80C22, (C2×C4○D4).12C2, SmallGroup(64,255)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C8.C22
C1C2C4C2×C4C22×C4C22×Q8 — C2×C8.C22
C1C2C4 — C2×C8.C22
C1C22C22×C4 — C2×C8.C22
C1C2C2C4 — C2×C8.C22

Generators and relations for C2×C8.C22
 G = < a,b,c,d | a2=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 185 in 129 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×6], Q8 [×7], C23, C23, C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C4○D4 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C2×C8.C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], C22×D4, C2×C8.C22

Character table of C2×C8.C22

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D
 size 1111224422224444444444
ρ11111111111111111111111    trivial
ρ21-11-11-1-11-1-1111-1-1-111-111-1    linear of order 2
ρ31-11-11-1-11-1-111-111-11-11-1-11    linear of order 2
ρ4111111111111-1-1-111-1-1-1-1-1    linear of order 2
ρ51111-1-1111-1-1111-1-1-1-1-1-111    linear of order 2
ρ61-11-1-11-11-11-111-111-1-11-11-1    linear of order 2
ρ71-11-1-11-11-11-11-11-11-11-11-11    linear of order 2
ρ81111-1-1111-1-11-1-11-1-1111-1-1    linear of order 2
ρ91-11-11-11-1-1-111-1111-1-1-111-1    linear of order 2
ρ10111111-1-11111-1-1-1-1-1-11111    linear of order 2
ρ11111111-1-11111111-1-11-1-1-1-1    linear of order 2
ρ121-11-11-11-1-1-1111-1-11-111-1-11    linear of order 2
ρ131-11-1-111-1-11-11-11-1-1111-11-1    linear of order 2
ρ141111-1-1-1-11-1-11-1-11111-1-111    linear of order 2
ρ151111-1-1-1-11-1-1111-111-111-1-1    linear of order 2
ρ161-11-1-111-1-11-111-11-11-1-11-11    linear of order 2
ρ172222-2-200-222-20000000000    orthogonal lifted from D4
ρ182-22-2-22002-22-20000000000    orthogonal lifted from D4
ρ1922222200-2-2-2-20000000000    orthogonal lifted from D4
ρ202-22-22-20022-2-20000000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-4-44000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C2×C8.C22
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(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 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 31)(18 28)(19 25)(20 30)(21 27)(22 32)(23 29)(24 26)

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

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

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

C2×C8.C22 is a maximal subgroup of
C8.C22⋊C4  M4(2).46D4  C42.6D4  M4(2).49D4  C4210D4  C42.130D4  M4(2)⋊4D4  M4(2).D4  M4(2).5D4  M4(2).6D4  M4(2).9D4  M4(2).11D4  C42.276C23  C24.178D4  C42.13C23  C42.212D4  C42.445D4  C42.16C23  C42.17C23  M4(2)⋊15D4  M4(2)⋊17D4  M4(2)⋊8D4  M4(2)⋊9D4  M4(2)⋊10D4  M4(2).20D4  Q164D4  C4.C25
 D4.pD4⋊C2: C24.104D4  C24.106D4  D4.(C2×D4)  Q8.(C2×D4)  (C2×Q8)⋊17D4  C42.446D4  C42.19C23  M4(2).38D4 ...
C2×C8.C22 is a maximal quotient of
C24.178D4  C24.106D4  C42.212D4  C42.445D4  M4(2)⋊15D4  C42.220D4  C42.448D4  C24.183D4  C24.118D4  C42.451D4  C42.226D4  C42.228D4  C42.230D4  C42.231D4  C42.234D4  C42.235D4  C42.241D4  C42.243D4  M4(2)⋊8D4  M4(2)⋊5Q8  C42.256D4  C42.258D4  C42.259D4  C42.262D4  C24.123D4  C24.126D4  C24.128D4  C24.129D4  C42.264D4  C42.267D4  C42.268D4  C42.273D4  C42.274D4  C42.276D4  C42.278D4  C42.281D4  C42.283D4  C42.288D4  C42.289D4  C42.290D4  C42.291D4  C42.296D4  C42.300D4  C42.302D4  C42.303D4  SD166D4  Q169D4  SD163D4  Q165D4  C42.47C23  C42.49C23  C42.51C23  C42.55C23  C42.477C23  C42.478C23  C42.480C23  C42.482C23  C42.58C23  C42.60C23  C42.497C23  C42.498C23  C42.510C23  C42.513C23  C42.515C23  C42.516C23  SD16⋊Q8  Q164Q8  C42.73C23  C42.75C23

Matrix representation of C2×C8.C22 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
1020000
970000
0051200
005500
00512125
00551212
,
100000
7160000
001000
0001600
0010160
0001601
,
100000
010000
0016020
0001602
000010
000001

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,9,0,0,0,0,2,7,0,0,0,0,0,0,5,5,5,5,0,0,12,5,12,5,0,0,0,0,12,12,0,0,0,0,5,12],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,16,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,2,0,1,0,0,0,0,2,0,1] >;

C2×C8.C22 in GAP, Magma, Sage, TeX

C_2\times C_8.C_2^2
% in TeX

G:=Group("C2xC8.C2^2");
// GroupNames label

G:=SmallGroup(64,255);
// by ID

G=gap.SmallGroup(64,255);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,217,199,650,1444,730,88]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

Export

Character table of C2×C8.C22 in TeX

׿
×
𝔽