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G = C2≀C22order 64 = 26

Wreath product of C2 by C22

p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: C2C22, C23⋊D4, C241C22, C23.3C23, 2+ 1+42C2, (C2×C4)⋊D4, C23⋊C43C2, C22≀C21C2, C2.18C22≀C2, C22⋊C41C22, (C2×D4).9C22, C22.16(C2×D4), 2-Sylow(A8), Hol(C2×C4), SmallGroup(64,138)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2≀C22
C1C2C22C23C2×D42+ 1+4 — C2≀C22
C1C2C23 — C2≀C22
C1C2C23 — C2≀C22
C1C2C23 — C2≀C22

Generators and relations for C2≀C22
 G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 225 in 99 conjugacy classes, 27 normal (6 characteristic)
C1, C2, C2 [×8], C4 [×6], C22 [×3], C22 [×18], C2×C4 [×3], C2×C4 [×6], D4 [×15], Q8, C23, C23 [×3], C23 [×6], C22⋊C4 [×3], C22⋊C4 [×3], C2×D4 [×3], C2×D4 [×6], C4○D4 [×3], C24, C23⋊C4 [×3], C22≀C2 [×3], 2+ 1+4, C2≀C22
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C2≀C22

Character table of C2≀C22

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F
 size 1122244444444888
ρ11111111111111111    trivial
ρ211111-1-1-11-11-1-11-11    linear of order 2
ρ3111111-1-1111-1-1-11-1    linear of order 2
ρ411111-1111-1111-1-1-1    linear of order 2
ρ5111111-11-11-1-111-1-1    linear of order 2
ρ611111-11-1-1-1-11-111-1    linear of order 2
ρ71111111-1-11-11-1-1-11    linear of order 2
ρ811111-1-11-1-1-1-11-111    linear of order 2
ρ922-2-220020000-2000    orthogonal lifted from D4
ρ10222-2-20-2000020000    orthogonal lifted from D4
ρ1122-2-2200-200002000    orthogonal lifted from D4
ρ1222-22-2000-20200000    orthogonal lifted from D4
ρ1322-22-200020-200000    orthogonal lifted from D4
ρ14222-2-2020000-20000    orthogonal lifted from D4
ρ154-4000-20002000000    orthogonal faithful
ρ164-40002000-2000000    orthogonal faithful

Permutation representations of C2≀C22
On 8 points - transitive group 8T29
Generators in S8
(1 2)(3 5)(4 8)(6 7)
(1 3)(2 5)(4 7)(6 8)
(1 6)(2 7)(3 8)(4 5)
(1 2 3 4)(5 6 7 8)
(2 4)(5 7)

G:=sub<Sym(8)| (1,2)(3,5)(4,8)(6,7), (1,3)(2,5)(4,7)(6,8), (1,6)(2,7)(3,8)(4,5), (1,2,3,4)(5,6,7,8), (2,4)(5,7)>;

G:=Group( (1,2)(3,5)(4,8)(6,7), (1,3)(2,5)(4,7)(6,8), (1,6)(2,7)(3,8)(4,5), (1,2,3,4)(5,6,7,8), (2,4)(5,7) );

G=PermutationGroup([(1,2),(3,5),(4,8),(6,7)], [(1,3),(2,5),(4,7),(6,8)], [(1,6),(2,7),(3,8),(4,5)], [(1,2,3,4),(5,6,7,8)], [(2,4),(5,7)])

G:=TransitiveGroup(8,29);

On 8 points - transitive group 8T31
Generators in S8
(1 6)(2 5)(3 7)(4 8)
(2 3)(5 7)
(1 4)(2 3)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)
(6 8)

G:=sub<Sym(8)| (1,6)(2,5)(3,7)(4,8), (2,3)(5,7), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8), (6,8)>;

G:=Group( (1,6)(2,5)(3,7)(4,8), (2,3)(5,7), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8), (6,8) );

G=PermutationGroup([(1,6),(2,5),(3,7),(4,8)], [(2,3),(5,7)], [(1,4),(2,3),(5,7),(6,8)], [(1,2),(3,4),(5,6,7,8)], [(6,8)])

G:=TransitiveGroup(8,31);

On 16 points - transitive group 16T127
Generators in S16
(1 12)(2 11)(3 15)(4 14)(5 13)(6 16)(7 10)(8 9)
(2 8)(4 6)(9 11)(14 16)
(1 7)(2 8)(3 5)(4 6)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 6)(3 7)(4 8)(9 14)(10 13)(11 16)(12 15)

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

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

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

G:=TransitiveGroup(16,127);

On 16 points - transitive group 16T128
Generators in S16
(1 7)(2 14)(3 13)(4 8)(5 9)(6 12)(10 16)(11 15)
(1 9)(2 4)(3 11)(5 7)(6 16)(8 14)(10 12)(13 15)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 15)(6 14)(7 13)(8 16)(10 12)

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

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

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

G:=TransitiveGroup(16,128);

On 16 points - transitive group 16T129
Generators in S16
(1 11)(2 12)(3 6)(4 7)(5 16)(8 15)(9 13)(10 14)
(1 15)(3 13)(6 9)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11)(2 10)(3 9)(4 12)(5 14)(6 13)(7 16)(8 15)

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

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

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

G:=TransitiveGroup(16,129);

On 16 points - transitive group 16T147
Generators in S16
(2 8)(3 13)(4 11)(5 10)(6 14)(9 16)
(1 12)(2 8)(3 10)(4 6)(5 13)(7 15)(9 16)(11 14)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 16)(14 15)

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

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

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

G:=TransitiveGroup(16,147);

On 16 points - transitive group 16T149
Generators in S16
(1 8)(2 7)(3 13)(4 16)(5 10)(6 9)(11 15)(12 14)
(1 3)(2 10)(4 12)(5 7)(6 15)(8 13)(9 11)(14 16)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11)(2 10)(3 9)(4 12)(5 16)(6 15)(7 14)(8 13)

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

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

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

G:=TransitiveGroup(16,149);

On 16 points - transitive group 16T150
Generators in S16
(1 13)(2 10)(3 16)(4 11)(5 15)(6 12)(7 14)(8 9)
(1 4)(2 7)(3 6)(5 8)(9 15)(10 14)(11 13)(12 16)
(1 5)(2 6)(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)
(1 6)(2 5)(3 4)(7 8)(9 12)(10 11)(13 14)(15 16)

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

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

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

G:=TransitiveGroup(16,150);

C2≀C22 is a maximal subgroup of
C2≀A4  C23⋊S4
 C24⋊D2p: D4≀C2  C425D4  C24⋊C23  C246D6  C24⋊D6  C242D10  C24⋊D14 ...
 (C22×C2p)⋊D4: C424D4  C426D4  C23.7C24  C23.9C24  C23⋊D12  2+ 1+47S3  C23⋊D20  2+ 1+42D5 ...
C2≀C22 is a maximal quotient of
C23⋊SD16  C4⋊C4.D4  (C2×C4)⋊D8  (C2×C4)⋊SD16  C24.9D4  C232SD16  C23⋊Q16  C4⋊C4.6D4  Q8⋊D4⋊C2  (C2×C4)⋊Q16  C24.12D4  2+ 1+42C4  C24.22D4  C242Q8  C24.33D4  C24.182C23  C42.13D4  C42.14D4  C42.15D4  C42.16D4  C42.17D4  Q8≀C2
 C23⋊D4p: C23⋊D8  C23⋊D12  C23⋊D20  C23⋊D28 ...
 C24⋊D2p: C24⋊D4  D4≀C2  C425D4  C246D6  C242D10  C24⋊D14 ...
 (C22×C2p)⋊D4: C25.C22  C424D4  C426D4  2+ 1+47S3  2+ 1+42D5  2+ 1+42D7 ...

Polynomial with Galois group C2≀C22 over ℚ
actionf(x)Disc(f)
8T29x8-x6+x2+1214·74
8T31x8-4x7-4x6+18x5+8x4-20x3-3x2+6x-1212·54·761

Matrix representation of C2≀C22 in GL4(ℤ) generated by

1000
0-100
0010
000-1
,
-1000
0-100
0010
0001
,
-1000
0-100
00-10
000-1
,
0001
0010
1000
0100
,
1000
0100
0001
0010
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C2≀C22 in GAP, Magma, Sage, TeX

C_2\wr C_2^2
% in TeX

G:=Group("C2wrC2^2");
// GroupNames label

G:=SmallGroup(64,138);
// by ID

G=gap.SmallGroup(64,138);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,255,730]);
// Polycyclic

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

Export

Character table of C2≀C22 in TeX

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