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G = C2xC8:C22  order 64 = 26

Direct product of C2 and C8:C22

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

Aliases: C2xC8:C22, C8:C23, D4:2C23, D8:3C22, C4.5C24, Q8:2C23, C23.50D4, SD16:1C22, M4(2):3C22, (C2xD8):11C2, (C2xC8):2C22, C4.64(C2xD4), (C2xSD16):4C2, (C2xC4).135D4, C4oD4:4C22, (C2xD4):15C22, (C22xD4):11C2, (C2xM4(2)):3C2, (C2xQ8):14C22, C2.27(C22xD4), C22.23(C2xD4), (C2xC4).139C23, (C22xC4).79C22, (C2xC4oD4):11C2, SmallGroup(64,254)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2xC8:C22
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C22xD4 — C2xC8:C22
Lower central
C1 — C2 — C4 — C2xC8:C22
Upper central
C1 — C22 — C22xC4 — C2xC8:C22
Jennings
C1 — C2 — C2 — C4 — C2xC8:C22

Generators and relations for C2xC8: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, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C2xC8, M4(2), D8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C24, C2xM4(2), C2xD8, C2xSD16, C8:C22, C22xD4, C2xC4oD4, C2xC8:C22
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C8:C22, C22xD4, C2xC8:C22

Character table of C2xC8: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 C2xC8: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);
 

C2xC8:C22 is a maximal subgroup of
 C8:C22:C4  M4(2).47D4  C42.5D4  M4(2).48D4  C42:9D4  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  C4oD4:D4  (C2xQ8):16D4  (C2xD4):21D4  C42.12C23  C42.211D4  C42.444D4  C42.446D4  C42.14C23  C42.15C23  C42.18C23  M4(2).37D4  D8:10D4  D8:5D4  D8:C23
 C8:pD4:C2: M4(2):14D4  M4(2):16D4  M4(2):7D4  M4(2):9D4  M4(2):10D4  M4(2):11D4  D8:9D4  SD16:D4 ...
C2xC8: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  D8:4Q8  SD16:2Q8
 C8:pD4:C2: M4(2):14D4  M4(2):7D4  C42.255D4  C42.257D4  C42.259D4  C42.261D4  C24.121D4  C24.125D4 ...

Matrix representation of C2xC8: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] >;
 

C2xC8: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 C2xC8:C22 in TeX

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