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G = C2×C4.D4order 64 = 26

Direct product of C2 and C4.D4

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

Aliases: C2×C4.D4, C24.2C4, M4(2)⋊8C22, (C2×D4).6C4, C4.44(C2×D4), (C2×C4).120D4, C23.4(C2×C4), (C2×C4).1C23, (C2×M4(2))⋊8C2, (C22×D4).5C2, C4.10(C22⋊C4), (C2×D4).44C22, C22.8(C22×C4), (C22×C4).31C22, C22.30(C22⋊C4), (C2×C4).20(C2×C4), C2.14(C2×C22⋊C4), SmallGroup(64,92)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C4.D4
C1C2C4C2×C4C22×C4C22×D4 — C2×C4.D4
C1C2C22 — C2×C4.D4
C1C22C22×C4 — C2×C4.D4
C1C2C2C2×C4 — C2×C4.D4

Generators and relations for C2×C4.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 [×2], C2 [×6], C4 [×4], C22 [×3], C22 [×18], C8 [×4], C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×4], C23 [×8], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C2×D4 [×4], C2×D4 [×4], C24 [×2], C4.D4 [×4], C2×M4(2) [×2], C22×D4, C2×C4.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×C4.D4

Character table of C2×C4.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 C2×C4.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);

C2×C4.D4 is a maximal subgroup of
C24.5D4  C25.C4  (C23×C4).C4  2+ 1+43C4  C42.96D4  C24.6(C2×C4)  C24.21D4  C24.23D4  C24.24D4  M4(2)⋊20D4  M4(2).47D4  M4(2).48D4  C42⋊D4  M4(2)⋊21D4  C4.D43C4  M4(2)⋊12D4  C42.115D4  M4(2).31D4  M4(2).32D4  C429D4  C4210D4  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  (C2×C8).D4  C24.36D4  M4(2).24C23  M4(2)⋊C23  M4(2).37D4  D5⋊(C4.D4)
C2×C4.D4 is a maximal quotient of
C42.393D4  C25.3C4  C42.43D4  C42.395D4  C24.(C2×C4)  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  (C22×C4).275D4  M4(2)⋊20D4  M4(2)⋊12D4  M4(2)⋊8Q8  D5⋊(C4.D4)

Matrix representation of C2×C4.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] >;

C2×C4.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

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Character table of C2×C4.D4 in TeX

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