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G = C2xC4.D4  order 64 = 26

Direct product of C2 and C4.D4

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

Aliases: C2xC4.D4, C24.2C4, M4(2):8C22, (C2xD4).6C4, C4.44(C2xD4), (C2xC4).120D4, C23.4(C2xC4), (C2xC4).1C23, (C2xM4(2)):8C2, (C22xD4).5C2, C4.10(C22:C4), (C2xD4).44C22, C22.8(C22xC4), (C22xC4).31C22, C22.30(C22:C4), (C2xC4).20(C2xC4), C2.14(C2xC22:C4), SmallGroup(64,92)

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

Derived series
C1 — C22 — C2xC4.D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C22xD4 — C2xC4.D4
Lower central
C1 — C2 — C22 — C2xC4.D4
Upper central
C1 — C22 — C22xC4 — C2xC4.D4
Jennings
C1 — C2 — C2 — C2xC4 — C2xC4.D4

Generators and relations for C2xC4.D4
 G = < a,b,c,d | a2=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 185 in 93 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2xC4, C2xC4, D4, C23, C23, C23, C2xC8, M4(2), M4(2), C22xC4, C2xD4, C2xD4, C24, C4.D4, C2xM4(2), C22xD4, C2xC4.D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4.D4, C2xC22:C4, C2xC4.D4

Character table of C2xC4.D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D8A8B8C8D8E8F8G8H
 size 1111224444222244444444
ρ11111111111111111111111    trivial
ρ21-11-1-11-11-11-1-111-1-11-11-111    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41-11-1-11-11-11-1-11111-11-11-1-1    linear of order 2
ρ51-11-1-111-11-1-1-111-11-111-11-1    linear of order 2
ρ6111111-1-1-1-111111-1-1-1111-1    linear of order 2
ρ71-11-1-111-11-1-1-1111-11-1-11-11    linear of order 2
ρ8111111-1-1-1-11111-1111-1-1-11    linear of order 2
ρ9111111-111-1-1-1-1-1i-iii-i-ii-i    linear of order 4
ρ101-11-1-1111-1-111-1-1-iii-i-iii-i    linear of order 4
ρ11111111-111-1-1-1-1-1-ii-i-iii-ii    linear of order 4
ρ121-11-1-1111-1-111-1-1i-i-iii-i-ii    linear of order 4
ρ131-11-1-11-1-11111-1-1-i-i-ii-iiii    linear of order 4
ρ141111111-1-11-1-1-1-1ii-i-i-i-iii    linear of order 4
ρ151-11-1-11-1-11111-1-1iii-ii-i-i-i    linear of order 4
ρ161111111-1-11-1-1-1-1-i-iiiii-i-i    linear of order 4
ρ172-22-22-20000-22-2200000000    orthogonal lifted from D4
ρ182222-2-20000-222-200000000    orthogonal lifted from D4
ρ192-22-22-200002-22-200000000    orthogonal lifted from D4
ρ202222-2-200002-2-2200000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    orthogonal lifted from C4.D4
ρ224-4-44000000000000000000    orthogonal lifted from C4.D4

Permutation representations of C2xC4.D4
►On 16 points - transitive group 16T72
Generators in S16
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)

(1 13 5 9)(2 10 6 14)(3 15 7 11)(4 12 8 16)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 10 13 6 5 14 9 2)(3 16 15 4 7 12 11 8)
 
G:=sub<Sym(16)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,13,5,9)(2,10,6,14)(3,15,7,11)(4,12,8,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10,13,6,5,14,9,2)(3,16,15,4,7,12,11,8)>;
 
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,13,5,9)(2,10,6,14)(3,15,7,11)(4,12,8,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10,13,6,5,14,9,2)(3,16,15,4,7,12,11,8) );
 
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10)], [(1,13,5,9),(2,10,6,14),(3,15,7,11),(4,12,8,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,10,13,6,5,14,9,2),(3,16,15,4,7,12,11,8)]])
 
G:=TransitiveGroup(16,72);
 

►On 16 points - transitive group 16T99
Generators in S16
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)

(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 10 3 16 5 14 7 12)(2 11 8 13 6 15 4 9)
 
G:=sub<Sym(16)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10,3,16,5,14,7,12)(2,11,8,13,6,15,4,9)>;
 
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,3,5,7)(2,8,6,4)(9,11,13,15)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10,3,16,5,14,7,12)(2,11,8,13,6,15,4,9) );
 
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10)], [(1,3,5,7),(2,8,6,4),(9,11,13,15),(10,16,14,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,10,3,16,5,14,7,12),(2,11,8,13,6,15,4,9)]])
 
G:=TransitiveGroup(16,99);
 

C2xC4.D4 is a maximal subgroup of
 C24.5D4  C25.C4  (C23xC4).C4  2+ 1+4:3C4  C42.96D4  C24.6(C2xC4)  C24.21D4  C24.23D4  C24.24D4  M4(2):20D4  M4(2).47D4  M4(2).48D4  C42:D4  M4(2):21D4  C4.D4:3C4  M4(2):12D4  C42.115D4  M4(2).31D4  M4(2).32D4  C42:9D4  C42:10D4  M4(2).4D4  M4(2).5D4  M4(2):6D4  C4:C4.96D4  C4:C4.97D4  M4(2).8D4  M4(2).10D4  M4(2).12D4  C24.Q8  (C2xC8).D4  C24.36D4  M4(2).24C23  M4(2):C23  M4(2).37D4  D5:(C4.D4)
C2xC4.D4 is a maximal quotient of
 C42.393D4  C25.3C4  C42.43D4  C42.395D4  C24.(C2xC4)  C42.405D4  C42.407D4  C42.67D4  C42.70D4  C42.73D4  C42.411D4  C42.413D4  C42.415D4  C42.80D4  C42.82D4  C42.85D4  C42.87D4  C25.C4  C42.96D4  (C22xC4).275D4  M4(2):20D4  M4(2):12D4  M4(2):8Q8  D5:(C4.D4)

Matrix representation of C2xC4.D4 ►in GL6(Z)

-100000
0-10000
001000
000100
000010
000001
,
100000
010000
000100
00-1000
000001
0000-10
,
-1-10000
210000
000001
000010
001000
000-100
,
110000
0-10000
0000-10
00000-1
000-100
001000

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,2,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,-1,0,0] >;
 

C2xC4.D4 in GAP, Magma, Sage, TeX 

C_2\times C_4.D_4
 
% in TeX
 
G:=Group("C2xC4.D4");
 
// GroupNames label
 
G:=SmallGroup(64,92);
 
// by ID
 
G=gap.SmallGroup(64,92);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,730,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
 
// generators/relations
 

Export

Character table of C2xC4.D4 in TeX

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