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G = C22≀C2order 32 = 25

Wreath product of C22 by C2

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C22C2, C241C2, C222D4, C231C22, C22.10C23, (C2×D4)⋊1C2, C2.4(C2×D4), C22⋊C42C2, (C2×C4)⋊1C22, 2-Sylow(SO+(4,4)), SmallGroup(32,27)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22≀C2
C1C2C22C23C24 — C22≀C2
C1C22 — C22≀C2
C1C22 — C22≀C2
C1C22 — C22≀C2

Generators and relations for C22≀C2
 G = < a,b,c,d,e | a2=b2=c2=d2=e2=1, ab=ba, eae=ac=ca, ad=da, bc=cb, ebe=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 106 in 65 conjugacy classes, 26 normal (5 characteristic)
C1, C2 [×3], C2 [×7], C4 [×3], C22, C22 [×6], C22 [×17], C2×C4 [×3], D4 [×6], C23, C23 [×3], C23 [×6], C22⋊C4 [×3], C2×D4 [×3], C24, C22≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2

Character table of C22≀C2

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C
 size 11112222224444
ρ111111111111111    trivial
ρ21111-11-1-11-11-11-1    linear of order 2
ρ311111-11-1-1-111-1-1    linear of order 2
ρ41111-1-1-11-111-1-11    linear of order 2
ρ51111-11-1-11-1-11-11    linear of order 2
ρ61111111111-1-1-1-1    linear of order 2
ρ71111-1-1-11-11-111-1    linear of order 2
ρ811111-11-1-1-1-1-111    linear of order 2
ρ92-22-20200-200000    orthogonal lifted from D4
ρ1022-2-2000-2020000    orthogonal lifted from D4
ρ112-2-2220-20000000    orthogonal lifted from D4
ρ1222-2-200020-20000    orthogonal lifted from D4
ρ132-22-20-200200000    orthogonal lifted from D4
ρ142-2-22-2020000000    orthogonal lifted from D4

Permutation representations of C22≀C2
On 8 points - transitive group 8T18
Generators in S8
(1 2)(3 4)(5 6)(7 8)
(1 7)(2 8)(3 6)(4 5)
(1 8)(2 7)(3 6)(4 5)
(1 2)(3 5)(4 6)(7 8)
(1 6)(2 4)(3 8)(5 7)

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

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

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

G:=TransitiveGroup(8,18);

On 16 points - transitive group 16T39
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 7)(2 8)(3 15)(4 16)(5 13)(6 14)(9 12)(10 11)
(1 8)(2 7)(3 16)(4 15)(5 11)(6 12)(9 14)(10 13)
(1 3)(2 4)(5 14)(6 13)(7 15)(8 16)(9 11)(10 12)
(1 12)(2 5)(3 10)(4 14)(6 8)(7 11)(9 15)(13 16)

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

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

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

G:=TransitiveGroup(16,39);

On 16 points - transitive group 16T46
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 10)(2 9)(3 7)(4 8)(5 13)(6 14)(11 16)(12 15)
(1 16)(2 15)(3 14)(4 13)(5 8)(6 7)(9 12)(10 11)
(1 14)(2 13)(3 16)(4 15)(5 9)(6 10)(7 11)(8 12)
(1 16)(3 14)(5 9)(6 11)(7 10)(8 12)

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

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

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

G:=TransitiveGroup(16,46);

C22≀C2 is a maximal subgroup of
C22.19C24  C22.29C24  C22.32C24  D42  C22.45C24  C22.54C24  C24⋊C22  C24⋊C6  C22⋊S4  C24⋊D5  D6≀C2
 C23⋊D2p: C2≀C22  C233D4  C232D6  C23⋊D10  C23⋊D14  C23⋊D22  C23⋊D26 ...
 D2p⋊D4: D45D4  D6⋊D4  C22⋊D20  C22⋊D28  C22⋊D44  C22⋊D52 ...
 C23.D2p: C2≀C4  C244S3  A4⋊D4  C242D5  C24⋊D7  C24⋊D11  C24⋊D13 ...
C22≀C2 is a maximal quotient of
C23⋊Q8  C23.78C23  D4.8D4  D4.10D4  D6≀C2
 C23⋊D2p: C232D4  C2≀C22  C232D6  C23⋊D10  C23⋊D14  C23⋊D22  C23⋊D26 ...
 C22⋊D4p: C22⋊D8  D6⋊D4  C22⋊D20  C22⋊D28  C22⋊D44  C22⋊D52 ...
 C23.D2p: C243C4  C23.8Q8  C23.23D4  C23.10D4  Q8⋊D4  D4⋊D4  C22⋊SD16  C22⋊Q16 ...

Polynomial with Galois group C22≀C2 over ℚ
actionf(x)Disc(f)
8T18x8-x6-x4-x2+128·32·134

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

-1000
0100
00-10
000-1
,
-1000
0100
0010
000-1
,
-1000
0-100
0010
0001
,
-1000
0-100
00-10
000-1
,
0100
1000
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],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C22≀C2 in GAP, Magma, Sage, TeX

C_2^2\wr C_2
% in TeX

G:=Group("C2^2wrC2");
// GroupNames label

G:=SmallGroup(32,27);
// by ID

G=gap.SmallGroup(32,27);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,101,302]);
// Polycyclic

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

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Character table of C22≀C2 in TeX

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