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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, C8⋊C23, D42C23, D83C22, C4.5C24, Q82C23, C23.50D4, SD161C22, M4(2)⋊3C22, (C2×D8)⋊11C2, (C2×C8)⋊2C22, C4.64(C2×D4), (C2×SD16)⋊4C2, (C2×C4).135D4, C4○D44C22, (C2×D4)⋊15C22, (C22×D4)⋊11C2, (C2×M4(2))⋊3C2, (C2×Q8)⋊14C22, C2.27(C22×D4), C22.23(C2×D4), (C2×C4).139C23, (C22×C4).79C22, (C2×C4○D4)⋊11C2, SmallGroup(64,254)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×C8⋊C22
C1C2C4C2×C4C22×C4C22×D4 — 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, cd=dc >

Subgroups: 265 in 149 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, 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 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F8A8B8C8D
 size 1111224444442222444444
ρ11111111111111111111111    trivial
ρ21-11-1-111-11-1-11-11-11-11-1-111    linear of order 2
ρ31-11-1-11-111-11-1-11-11-1111-1-1    linear of order 2
ρ4111111-1-111-1-1111111-1-1-1-1    linear of order 2
ρ51-11-1-11-11-111-1-11-111-1-1-111    linear of order 2
ρ6111111-1-1-1-1-1-11111-1-11111    linear of order 2
ρ711111111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ81-11-1-111-1-11-11-11-111-111-1-1    linear of order 2
ρ91-11-11-111-11-1-111-1-1-11-111-1    linear of order 2
ρ101111-1-11-1-1-11-1-111-1111-11-1    linear of order 2
ρ111111-1-1-11-1-1-11-111-111-11-11    linear of order 2
ρ121-11-11-1-1-1-111111-1-1-111-1-11    linear of order 2
ρ131111-1-1-1111-11-111-1-1-11-11-1    linear of order 2
ρ141-11-11-1-1-11-11111-1-11-1-111-1    linear of order 2
ρ151-11-11-1111-1-1-111-1-11-11-1-11    linear of order 2
ρ161111-1-11-1111-1-111-1-1-1-11-11    linear of order 2
ρ17222222000000-2-2-2-2000000    orthogonal lifted from D4
ρ182222-2-20000002-2-22000000    orthogonal lifted from D4
ρ192-22-2-220000002-22-2000000    orthogonal lifted from D4
ρ202-22-22-2000000-2-222000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44000000000000000000    orthogonal lifted from C8⋊C22

Permutation representations of C2×C8⋊C22
On 16 points - transitive group 16T89
Generators in S16
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)
(1 5)(3 7)(9 13)(11 15)

G:=sub<Sym(16)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16), (1,5)(3,7)(9,13)(11,15)>;

G:=Group( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16), (1,5)(3,7)(9,13)(11,15) );

G=PermutationGroup([(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16)], [(1,5),(3,7),(9,13),(11,15)])

G:=TransitiveGroup(16,89);

C2×C8⋊C22 is a maximal subgroup of
C8⋊C22⋊C4  M4(2).47D4  C42.5D4  M4(2).48D4  C429D4  C42.129D4  M4(2)⋊D4  M4(2)⋊5D4  M4(2).4D4  M4(2).5D4  M4(2).8D4  M4(2).10D4  C42.275C23  C24.177D4  C24.104D4  C24.105D4  C4○D4⋊D4  (C2×Q8)⋊16D4  (C2×D4)⋊21D4  C42.12C23  C42.211D4  C42.444D4  C42.446D4  C42.14C23  C42.15C23  C42.18C23  M4(2).37D4  D810D4  D85D4  D8⋊C23
 C8pD4⋊C2: M4(2)⋊14D4  M4(2)⋊16D4  M4(2)⋊7D4  M4(2)⋊9D4  M4(2)⋊10D4  M4(2)⋊11D4  D89D4  SD16⋊D4 ...
C2×C8⋊C22 is a maximal quotient of
C24.177D4  C24.105D4  C42.211D4  C42.444D4  C42.219D4  C42.448D4  C24.183D4  C24.117D4  C42.225D4  C42.450D4  C42.227D4  C42.228D4  C42.230D4  C42.232D4  C42.233D4  C42.240D4  C42.243D4  M4(2)⋊5Q8  C24.126D4  C42.263D4  C42.279D4  C42.280D4  C42.282D4  C42.286D4  C42.287D4  C42.290D4  C42.291D4  C42.302D4  C42.45C23  C42.473C23  C42.479C23  C42.57C23  C42.494C23  C42.507C23  C42.508C23  C42.509C23  C42.514C23  D84Q8  SD162Q8
 C8pD4⋊C2: M4(2)⋊14D4  M4(2)⋊7D4  C42.255D4  C42.257D4  C42.259D4  C42.261D4  C24.121D4  C24.125D4 ...

Matrix representation of C2×C8⋊C22 in GL6(ℤ)

-100000
0-10000
001000
000100
000010
000001
,
-1-20000
110000
001020
001011
000-1-10
0000-10
,
-100000
110000
001000
001-100
00-100-1
00-10-10
,
-100000
0-10000
001000
000100
00-10-10
00-100-1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0,2,1,-1,-1,0,0,0,1,0,0],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,-1,-1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

C2×C8⋊C22 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes C_2^2
% in TeX

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

G:=SmallGroup(64,254);
// by ID

G=gap.SmallGroup(64,254);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,217,650,1444,730,88]);
// 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,c*d=d*c>;
// generators/relations

Export

Character table of C2×C8⋊C22 in TeX

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