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G = C2×C2≀C4order 128 = 27

Direct product of C2 and C2≀C4

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2×C2≀C4, C252C4, C24.34D4, C245(C2×C4), C22≀C25C4, C23.1(C2×D4), (C2×D4).123D4, C23⋊C42C22, (C22×C4).88D4, (C2×D4).12C23, C4.D417C22, C23.51(C22×C4), C22≀C2.19C22, C22.48(C23⋊C4), (C22×D4).98C22, C23.201(C22⋊C4), (C2×C4).1(C2×D4), C22⋊C41(C2×C4), (C2×C22⋊C4)⋊7C4, (C2×C23⋊C4)⋊11C2, C2.30(C2×C23⋊C4), (C2×C22≀C2).3C2, (C2×D4).121(C2×C4), (C2×C4.D4)⋊26C2, (C2×C4).23(C22⋊C4), C22.54(C2×C22⋊C4), SmallGroup(128,850)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2×C2≀C4
C1C2C22C23C2×D4C22×D4C2×C22≀C2 — C2×C2≀C4
C1C2C22C23 — C2×C2≀C4
C1C22C23C22×D4 — C2×C2≀C4
C1C2C22C2×D4 — C2×C2≀C4

Generators and relations for C2×C2≀C4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 660 in 219 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×6], C22 [×3], C22 [×49], C8 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×12], C23 [×5], C23 [×46], C22⋊C4 [×2], C22⋊C4 [×8], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C24 [×2], C24 [×2], C24 [×8], C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.D4, C2×C22⋊C4, C2×C22⋊C4 [×2], C22≀C2 [×4], C22≀C2 [×2], C2×M4(2), C22×D4, C22×D4, C25, C2≀C4 [×4], C2×C23⋊C4, C2×C4.D4, C2×C22≀C2, C2×C2≀C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2≀C4 [×2], C2×C23⋊C4, C2×C2≀C4

Character table of C2×C2≀C4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H8A8B8C8D
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ21-11-11-11-1-1111-1-1-11-11-111-111-1-1    linear of order 2
ρ3111111-11-1-111-1111-1-1-1-1-1-11111    linear of order 2
ρ41-11-11-1-1-11-1111-1-111-11-1-1111-1-1    linear of order 2
ρ51-11-11-1-1-11-1111-1-11-1111-1-1-1-111    linear of order 2
ρ6111111-11-1-111-111111-11-11-1-1-1-1    linear of order 2
ρ71-11-11-11-1-1111-1-1-111-1-1-111-1-111    linear of order 2
ρ81111111111111111-1-11-11-1-1-1-1-1    linear of order 2
ρ9111111-11-1-1-11-1-1-1-1i-i1i1-ii-i-ii    linear of order 4
ρ101-11-11-1-1-11-1-11111-1-i-i-1i1ii-ii-i    linear of order 4
ρ111-11-11-1-1-11-1-11111-1ii-1-i1-i-ii-ii    linear of order 4
ρ12111111-11-1-1-11-1-1-1-1-ii1-i1i-iii-i    linear of order 4
ρ131-11-11-11-1-11-11-111-1-i-i1i-1i-ii-ii    linear of order 4
ρ141111111111-111-1-1-1i-i-1i-1-i-iii-i    linear of order 4
ρ151111111111-111-1-1-1-ii-1-i-1ii-i-ii    linear of order 4
ρ161-11-11-11-1-11-11-111-1ii1-i-1-ii-ii-i    linear of order 4
ρ172-22-22-20200-2-202-220000000000    orthogonal lifted from D4
ρ182-22-22-202002-20-22-20000000000    orthogonal lifted from D4
ρ192222220-2002-202-2-20000000000    orthogonal lifted from D4
ρ202222220-200-2-20-2220000000000    orthogonal lifted from D4
ρ214-4-4400-202200-20000000000000    orthogonal lifted from C2≀C4
ρ2244-4-400-20-220020000000000000    orthogonal lifted from C2≀C4
ρ234-44-4-4400000000000000000000    orthogonal lifted from C23⋊C4
ρ244444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ254-4-440020-2-20020000000000000    orthogonal lifted from C2≀C4
ρ2644-4-400202-200-20000000000000    orthogonal lifted from C2≀C4

Permutation representations of C2×C2≀C4
On 16 points - transitive group 16T227
Generators in S16
(1 15)(2 16)(3 13)(4 14)(5 7)(6 8)(9 11)(10 12)
(1 3)(2 7)(4 11)(5 16)(6 10)(8 12)(9 14)(13 15)
(1 8)(2 11)(3 12)(4 7)(5 14)(6 15)(9 16)(10 13)
(2 14)(4 16)(5 11)(7 9)
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

G:=Group( (1,15)(2,16)(3,13)(4,14)(5,7)(6,8)(9,11)(10,12), (1,3)(2,7)(4,11)(5,16)(6,10)(8,12)(9,14)(13,15), (1,8)(2,11)(3,12)(4,7)(5,14)(6,15)(9,16)(10,13), (2,14)(4,16)(5,11)(7,9), (1,13)(2,14)(3,15)(4,16)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,7),(6,8),(9,11),(10,12)], [(1,3),(2,7),(4,11),(5,16),(6,10),(8,12),(9,14),(13,15)], [(1,8),(2,11),(3,12),(4,7),(5,14),(6,15),(9,16),(10,13)], [(2,14),(4,16),(5,11),(7,9)], [(1,13),(2,14),(3,15),(4,16),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,227);

On 16 points - transitive group 16T259
Generators in S16
(1 10)(2 11)(3 12)(4 9)(5 16)(6 13)(7 14)(8 15)
(1 10)(4 7)(8 15)(9 14)
(1 15)(2 11)(4 7)(5 16)(8 10)(9 14)
(1 10)(2 5)(3 12)(4 7)(6 13)(8 15)(9 14)(11 16)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

G:=Group( (1,10)(2,11)(3,12)(4,9)(5,16)(6,13)(7,14)(8,15), (1,10)(4,7)(8,15)(9,14), (1,15)(2,11)(4,7)(5,16)(8,10)(9,14), (1,10)(2,5)(3,12)(4,7)(6,13)(8,15)(9,14)(11,16), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,10),(2,11),(3,12),(4,9),(5,16),(6,13),(7,14),(8,15)], [(1,10),(4,7),(8,15),(9,14)], [(1,15),(2,11),(4,7),(5,16),(8,10),(9,14)], [(1,10),(2,5),(3,12),(4,7),(6,13),(8,15),(9,14),(11,16)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,259);

On 16 points - transitive group 16T261
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(2 15)(7 12)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

G:=Group( (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (2,15)(7,12), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(2,15),(7,12)], [(2,15),(3,16),(7,12),(8,9)], [(2,15),(4,13),(5,10),(7,12)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,261);

On 16 points - transitive group 16T273
Generators in S16
(1 11)(2 12)(3 9)(4 10)(5 15)(6 16)(7 13)(8 14)
(1 5)(2 16)(3 7)(6 12)(9 13)(11 15)
(1 5)(2 12)(3 9)(4 8)(6 16)(7 13)(10 14)(11 15)
(2 16)(4 14)(6 12)(8 10)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

G:=Group( (1,11)(2,12)(3,9)(4,10)(5,15)(6,16)(7,13)(8,14), (1,5)(2,16)(3,7)(6,12)(9,13)(11,15), (1,5)(2,12)(3,9)(4,8)(6,16)(7,13)(10,14)(11,15), (2,16)(4,14)(6,12)(8,10), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,11),(2,12),(3,9),(4,10),(5,15),(6,16),(7,13),(8,14)], [(1,5),(2,16),(3,7),(6,12),(9,13),(11,15)], [(1,5),(2,12),(3,9),(4,8),(6,16),(7,13),(10,14),(11,15)], [(2,16),(4,14),(6,12),(8,10)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,273);

On 16 points - transitive group 16T283
Generators in S16
(1 7)(2 8)(3 6)(4 5)(9 14)(10 15)(11 16)(12 13)
(1 6)(2 12)(3 7)(4 15)(5 10)(8 13)(9 11)(14 16)
(1 11)(2 10)(3 14)(4 13)(5 12)(6 9)(7 16)(8 15)
(2 5)(4 8)(10 12)(13 15)
(1 6)(2 5)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

G:=Group( (1,7)(2,8)(3,6)(4,5)(9,14)(10,15)(11,16)(12,13), (1,6)(2,12)(3,7)(4,15)(5,10)(8,13)(9,11)(14,16), (1,11)(2,10)(3,14)(4,13)(5,12)(6,9)(7,16)(8,15), (2,5)(4,8)(10,12)(13,15), (1,6)(2,5)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,7),(2,8),(3,6),(4,5),(9,14),(10,15),(11,16),(12,13)], [(1,6),(2,12),(3,7),(4,15),(5,10),(8,13),(9,11),(14,16)], [(1,11),(2,10),(3,14),(4,13),(5,12),(6,9),(7,16),(8,15)], [(2,5),(4,8),(10,12),(13,15)], [(1,6),(2,5),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,283);

Matrix representation of C2×C2≀C4 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
0160000
1600000
0016000
0001600
000010
0000016
,
1600000
0160000
0016000
000100
0000160
000001
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
400000
0130000
0000160
0000016
0001600
0016000

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,16,0,0] >;

C2×C2≀C4 in GAP, Magma, Sage, TeX

C_2\times C_2\wr C_4
% in TeX

G:=Group("C2xC2wrC4");
// GroupNames label

G:=SmallGroup(128,850);
// by ID

G=gap.SmallGroup(128,850);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,851,375,4037]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations

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Character table of C2×C2≀C4 in TeX

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