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G = C4○C2≀C4order 128 = 27

Central product of C4 and C2≀C4

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

Aliases: C4C2≀C4, C2≀C47C2, (C23×C4)⋊9C4, C22≀C26C4, C23.3(C2×D4), C42⋊C26C4, (C2×D4).125D4, C24.32(C2×C4), (C2×Q8).114D4, (C22×C4).90D4, C4.29(C23⋊C4), (C2×D4).14C23, C4(C23.D4), C23.D47C2, C22.D46C4, C23⋊C4.7C22, C23.53(C22×C4), C4.D4.8C22, C22≀C2.20C22, C23.19(C22⋊C4), C22.19C24.9C2, C23.C2312C2, C22.D4.24C22, M4(2).8C2214C2, (C2×C4).3(C2×D4), C22⋊C4.1(C2×C4), C2.32(C2×C23⋊C4), (C2×D4).123(C2×C4), (C22×C4).78(C2×C4), (C2×C4○D4).71C22, C22.56(C2×C22⋊C4), (C2×C4).364(C22⋊C4), SmallGroup(128,852)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4○C2≀C4
C1C2C22C23C2×D4C2×C4○D4C22.19C24 — C4○C2≀C4
C1C2C22C23 — C4○C2≀C4
C1C4C2×C4C2×C4○D4 — C4○C2≀C4
C1C2C22C2×D4 — C4○C2≀C4

Generators and relations for C4○C2≀C4
 G = < a,b,c,d,e,f | a4=b2=c2=d2=f4=1, e1=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=a2bcd, cd=dc, ce=ec, fcf-1=a2cd, de=ed, fdf-1=a2d, ef=fe >

Subgroups: 324 in 132 conjugacy classes, 42 normal (34 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×14], C8 [×2], C2×C4 [×4], C2×C4 [×16], D4 [×7], Q8, C23 [×3], C23 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C2×C8, M4(2) [×3], C22×C4 [×4], C22×C4 [×4], C2×D4 [×3], C2×D4 [×2], C2×Q8, C4○D4 [×2], C24, C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.10D4, C42⋊C2, C42⋊C2, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C22⋊Q8, C22.D4 [×2], C2×M4(2), C23×C4, C2×C4○D4, C2≀C4 [×2], C23.D4 [×2], C23.C23, M4(2).8C22, C22.19C24, C4○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×C23⋊C4, C4○C2≀C4

Character table of C4○C2≀C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11244444112444448888888888
ρ111111111111111111111111111    trivial
ρ2111-11111-1-1-1-1-1-11-1-111-11-1-1-111    linear of order 2
ρ31111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ4111-11111-1-1-1-1-1-11-11-1-111-111-1-1    linear of order 2
ρ5111-1-111-1-1-1-11-111-1-111-1-1111-1-1    linear of order 2
ρ61111-111-1111-11-1111111-1-1-1-1-1-1    linear of order 2
ρ71111-111-1111-11-111-1-1-1-1-1-11111    linear of order 2
ρ8111-1-111-1-1-1-11-111-11-1-11-11-1-111    linear of order 2
ρ9111-11-111111111-1-1-i-iii-1-1i-ii-i    linear of order 4
ρ1011111-111-1-1-1-1-1-1-11i-ii-i-11-iii-i    linear of order 4
ρ111111-1-11-1-1-1-11-11-11-ii-ii1-1-iii-i    linear of order 4
ρ12111-1-1-11-1111-11-1-1-1ii-i-i11i-ii-i    linear of order 4
ρ131111-1-11-1-1-1-11-11-11i-ii-i1-1i-i-ii    linear of order 4
ρ14111-1-1-11-1111-11-1-1-1-i-iii11-ii-ii    linear of order 4
ρ15111-11-111111111-1-1ii-i-i-1-1-ii-ii    linear of order 4
ρ1611111-111-1-1-1-1-1-1-11-ii-ii-11i-i-ii    linear of order 4
ρ17222-202-202220-20-220000000000    orthogonal lifted from D4
ρ18222-20-2-20-2-2-2020220000000000    orthogonal lifted from D4
ρ1922220-2-202220-202-20000000000    orthogonal lifted from D4
ρ20222202-20-2-2-2020-2-20000000000    orthogonal lifted from D4
ρ2144-40000044-4000000000000000    orthogonal lifted from C23⋊C4
ρ2244-400000-4-44000000000000000    orthogonal lifted from C23⋊C4
ρ234-400-20024i-4i02i0-2i000000000000    complex faithful
ρ244-400200-24i-4i0-2i02i000000000000    complex faithful
ρ254-400-2002-4i4i0-2i02i000000000000    complex faithful
ρ264-400200-2-4i4i02i0-2i000000000000    complex faithful

Permutation representations of C4○C2≀C4
On 16 points - transitive group 16T243
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 14 12 8)(2 15 9 5)(3 16 10 6)(4 13 11 7)

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

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

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

G:=TransitiveGroup(16,243);

On 16 points - transitive group 16T280
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 14)(6 15)(7 16)(8 13)
(1 11)(2 12)(3 9)(4 10)(5 14)(6 15)(7 16)(8 13)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 14)(2 15)(3 16)(4 13)(5 11 7 9)(6 12 8 10)

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

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

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

G:=TransitiveGroup(16,280);

On 16 points - transitive group 16T292
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(5 7)(6 8)
(5 7)(6 8)(9 11)(10 12)
(5 7)(6 8)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 5 11 16)(2 6 12 13)(3 7 9 14)(4 8 10 15)

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

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

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

G:=TransitiveGroup(16,292);

Matrix representation of C4○C2≀C4 in GL4(𝔽5) generated by

3000
0300
0030
0003
,
4000
0100
0010
0001
,
4000
0100
0040
0001
,
4000
0100
0010
0004
,
4000
0400
0040
0004
,
0300
0002
1000
0010
G:=sub<GL(4,GF(5))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,0,1,0,3,0,0,0,0,0,0,1,0,2,0,0] >;

C4○C2≀C4 in GAP, Magma, Sage, TeX

C_4\circ C_2\wr C_4
% in TeX

G:=Group("C4oC2wrC4");
// GroupNames label

G:=SmallGroup(128,852);
// by ID

G=gap.SmallGroup(128,852);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,352,1123,851,375,4037]);
// Polycyclic

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

Export

Character table of C4○C2≀C4 in TeX

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