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G = C2×C4≀C2order 64 = 26

Direct product of C2 and C4≀C2

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

Aliases: C2×C4≀C2, C23.39D4, C4215C22, M4(2)⋊9C22, C4C4≀C2, C4○D43C4, (C2×D4)⋊9C4, D45(C2×C4), (C2×Q8)⋊7C4, Q85(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

C1C4 — C2×C4≀C2
C1C2C4C2×C4C22×C4C2×C4○D4 — C2×C4≀C2
C1C2C4 — C2×C4≀C2
C1C2×C4C22×C4 — C2×C4≀C2
C1C2C2C2×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 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42, C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C2×C4≀C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], 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+43C4  2- 1+42C4  C42.102D4  C4≀C2⋊C4  C429(C2×C4)  M4(2).41D4  M4(2).42D4  C24.72D4  M4(2).43D4  C8.C22⋊C4  C8⋊C22⋊C4  (C2×C4)≀C2  C427D4  C42.426D4  M4(2).24D4  C42.427D4  C42.428D4  C42.107D4  C43⋊C2  C428D4  C42.326D4  C42.116D4  M4(2)⋊13D4  C429D4  C42.129D4  C4210D4  C42.130D4  M4(2)⋊D4  M4(2)⋊4D4  C42.8D4  M4(2)⋊6D4  M4(2).7D4  C4211D4  C4212D4  C42.131D4  2- 1+45C4  M4(2).51D4  C42.313C23  D5⋊C4≀C2
C2×C4≀C2 is a maximal quotient of
C42.455D4  C42.47D4  C42.400D4  C42.401D4  D44M4(2)  D45M4(2)  Q85M4(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  C4216Q8  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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