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G = C2×C4≀C2  order 64 = 26

Direct product of C2 and C4≀C2

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

Aliases: C2×C4≀C2, C23.39D4, C42⋊15C22, M4(2)⋊9C22, C4○C4≀C2, C4○D4⋊3C4, (C2×D4)⋊9C4, D4⋊5(C2×C4), (C2×Q8)⋊7C4, Q8⋊5(C2×C4), (C2×C42)⋊6C2, C4.69(C2×D4), (C2×C4).126D4, C4.7(C22×C4), (C2×C4).65C23, C4○D4.5C22, C22.10(C2×D4), C4.16(C22⋊C4), (C2×M4(2))⋊12C2, C22.34(C22⋊C4), (C22×C4).110C22, (C2×C4).46(C2×C4), (C2×C4○D4).6C2, C2.23(C2×C22⋊C4), SmallGroup(64,101)

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

Derived series
C1 — C4 — C2×C4≀C2
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C4≀C2
Lower central
C1 — C2 — C4 — C2×C4≀C2
Upper central
C1 — C2×C4 — C22×C4 — C2×C4≀C2
Jennings
C1 — C2 — C2 — C2×C4 — C2×C4≀C2

Generators and relations for C2×C4≀C2
 G = < a,b,c,d | a2=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 137 in 85 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C2×C4≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4≀C2, C2×C22⋊C4, C2×C4≀C2

Character table of C2×C4≀C2

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 1111224411112222222222444444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-111-1-111-1-1-1-1    linear of order 2
ρ3111111-1-11111-1-1-1-1-1-111-1-1-1-11111    linear of order 2
ρ4111111-1-111111111111111-1-1-1-1-1-1    linear of order 2
ρ51-11-11-1-11-111-1-1-1-111-1-11111-11-11-1    linear of order 2
ρ61-11-11-11-1-111-1-1-1-111-1-1111-11-11-11    linear of order 2
ρ71-11-11-1-11-111-1111-1-11-11-1-11-1-11-11    linear of order 2
ρ81-11-11-11-1-111-1111-1-11-11-1-1-111-11-1    linear of order 2
ρ91-11-1-11-111-1-11ii-iii-i-11-i-i-11ii-i-i    linear of order 4
ρ101-11-1-111-11-1-11ii-iii-i-11-i-i1-1-i-iii    linear of order 4
ρ111-11-1-11-111-1-11-i-ii-i-ii-11ii-11-i-iii    linear of order 4
ρ121-11-1-111-11-1-11-i-ii-i-ii-11ii1-1ii-i-i    linear of order 4
ρ131111-1-111-1-1-1-1ii-i-i-i-i11ii-1-1-iii-i    linear of order 4
ρ141111-1-1-1-1-1-1-1-1ii-i-i-i-i11ii11i-i-ii    linear of order 4
ρ151111-1-111-1-1-1-1-i-iiiii11-i-i-1-1i-i-ii    linear of order 4
ρ161111-1-1-1-1-1-1-1-1-i-iiiii11-i-i11-iii-i    linear of order 4
ρ172-22-2-2200-222-20000002-200000000    orthogonal lifted from D4
ρ1822222200-2-2-2-2000000-2-200000000    orthogonal lifted from D4
ρ192-22-22-2002-2-220000002-200000000    orthogonal lifted from D4
ρ202222-2-2002222000000-2-200000000    orthogonal lifted from D4
ρ212-2-220000-2i2i-2i2i1-i-1+i1+i1+i-1-i-1-i001-i-1+i000000    complex lifted from C4≀C2
ρ2222-2-20000-2i-2i2i2i1+i-1-i1-i-1+i1-i-1+i00-1-i1+i000000    complex lifted from C4≀C2
ρ2322-2-200002i2i-2i-2i1-i-1+i1+i-1-i1+i-1-i00-1+i1-i000000    complex lifted from C4≀C2
ρ242-2-2200002i-2i2i-2i-1-i1+i-1+i-1+i1-i1-i00-1-i1+i000000    complex lifted from C4≀C2
ρ2522-2-20000-2i-2i2i2i-1-i1+i-1+i1-i-1+i1-i001+i-1-i000000    complex lifted from C4≀C2
ρ262-2-220000-2i2i-2i2i-1+i1-i-1-i-1-i1+i1+i00-1+i1-i000000    complex lifted from C4≀C2
ρ2722-2-200002i2i-2i-2i-1+i1-i-1-i1+i-1-i1+i001-i-1+i000000    complex lifted from C4≀C2
ρ282-2-2200002i-2i2i-2i1+i-1-i1-i1-i-1+i-1+i001+i-1-i000000    complex lifted from C4≀C2

Permutation representations of C2×C4≀C2
►On 16 points - transitive group 16T111
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)

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

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

(1 5)(2 6)(3 7)(4 8)(9 14 11 16)(10 15 12 13)
 
G:=sub<Sym(16)| (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11), (1,5)(2,6)(3,7)(4,8)(9,14,11,16)(10,15,12,13)>;
 
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,15)(10,16)(11,13)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11), (1,5)(2,6)(3,7)(4,8)(9,14,11,16)(10,15,12,13) );
 
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,15),(10,16),(11,13),(12,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,10),(6,9),(7,12),(8,11)], [(1,5),(2,6),(3,7),(4,8),(9,14,11,16),(10,15,12,13)]])
 
G:=TransitiveGroup(16,111);
 

C2×C4≀C2 is a maximal subgroup of
 D4.C42  Q8.C42  D4.3C42  C24.66D4  2+ 1+4⋊3C4  2- 1+4⋊2C4  C42.102D4  C4≀C2⋊C4  C42⋊9(C2×C4)  M4(2).41D4  M4(2).42D4  C24.72D4  M4(2).43D4  C8.C22⋊C4  C8⋊C22⋊C4  (C2×C4)≀C2  C42⋊7D4  C42.426D4  M4(2).24D4  C42.427D4  C42.428D4  C42.107D4  C43⋊C2  C42⋊8D4  C42.326D4  C42.116D4  M4(2)⋊13D4  C42⋊9D4  C42.129D4  C42⋊10D4  C42.130D4  M4(2)⋊D4  M4(2)⋊4D4  C42.8D4  M4(2)⋊6D4  M4(2).7D4  C42⋊11D4  C42⋊12D4  C42.131D4  2- 1+4⋊5C4  M4(2).51D4  C42.313C23  D5⋊C4≀C2
C2×C4≀C2 is a maximal quotient of
 C42.455D4  C42.47D4  C42.400D4  C42.401D4  D4⋊4M4(2)  D4⋊5M4(2)  Q8⋊5M4(2)  C42.315D4  C42.316D4  C42.305D4  C42.375D4  C24.53D4  C42.403D4  C42.404D4  C24.150D4  C42.55D4  C42.56D4  C24.54D4  C24.55D4  C42.57D4  C42.66D4  C42.405D4  C42.406D4  C42.407D4  C42.408D4  C42.376D4  C42.67D4  C42.68D4  C42.69D4  C24.66D4  C42.102D4  C24.70D4  C24.72D4  (C2×C4)≀C2  C43⋊C2  M4(2)⋊13D4  M4(2)⋊7Q8  C42⋊16Q8  D5⋊C4≀C2

Matrix representation of C2×C4≀C2 ►in GL3(𝔽17) generated by

1600
0160
0016
,
100
040
0013
,
1600
0013
040
,
100
010
004
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,4,0,0,0,13],[16,0,0,0,0,4,0,13,0],[1,0,0,0,1,0,0,0,4] >;
 

C2×C4≀C2 in GAP, Magma, Sage, TeX 

C_2\times C_4\wr C_2
 
% in TeX
 
G:=Group("C2xC4wrC2");
 
// GroupNames label
 
G:=SmallGroup(64,101);
 
// by ID
 
G=gap.SmallGroup(64,101);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,489,117,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
 
// generators/relations
 

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Character table of C2×C4≀C2 in TeX

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