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G = C2×C22≀C2order 64 = 26

Direct product of C2 and C22≀C2

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

Aliases: C2×C22≀C2, C251C2, C235D4, C231C23, C246C22, C22.15C24, (C2×C4)⋊1C23, C223(C2×D4), (C2×D4)⋊9C22, (C22×D4)⋊3C2, C2.4(C22×D4), (C22×C4)⋊5C22, C22⋊C412C22, (C2×C22⋊C4)⋊8C2, SmallGroup(64,202)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C22≀C2
C1C2C22C23C24C25 — C2×C22≀C2
C1C22 — C2×C22≀C2
C1C23 — C2×C22≀C2
C1C22 — C2×C22≀C2

Generators and relations for C2×C22≀C2
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 569 in 331 conjugacy classes, 105 normal (6 characteristic)
C1, C2 [×7], C2 [×14], C4 [×6], C22, C22 [×18], C22 [×78], C2×C4 [×6], C2×C4 [×6], D4 [×24], C23, C23 [×20], C23 [×74], C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×12], C2×D4 [×12], C24, C24 [×7], C24 [×12], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C2×C22≀C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2

Character table of C2×C22≀C2

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O2P2Q2R2S2T2U4A4B4C4D4E4F
 size 1111111122222222222244444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-111-1-1-1-111-1-1-1-1-11-1111    linear of order 2
ρ31-1111-1-1-1-1-11-11-11-11-1111-1-111-1-11    linear of order 2
ρ41-1111-1-1-1111-1-11-111-1-1-1-1111-1-1-11    linear of order 2
ρ51-1111-1-1-1-1-11-11-11-11-111-111-1-111-1    linear of order 2
ρ611111111-1-111-1-1-1-111-1-1111-11-1-1-1    linear of order 2
ρ71-1111-1-1-1111-1-11-111-1-1-11-1-1-1111-1    linear of order 2
ρ811111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ9111111111-1-1-111-1-1-1-11-111-11-11-1-1    linear of order 2
ρ101-1111-1-1-11-1-11-111-1-11-11-11-111-11-1    linear of order 2
ρ1111111111-11-1-1-1-111-1-1-11-1-11111-1-1    linear of order 2
ρ121-1111-1-1-1-11-111-1-11-111-11-111-1-11-1    linear of order 2
ρ131-1111-1-1-11-1-11-111-1-11-111-11-1-11-11    linear of order 2
ρ14111111111-1-1-111-1-1-1-11-1-1-11-11-111    linear of order 2
ρ151-1111-1-1-1-11-111-1-11-111-1-11-1-111-11    linear of order 2
ρ1611111111-11-1-1-1-111-1-1-1111-1-1-1-111    linear of order 2
ρ172-2-2-2222-2-2000-2200002000000000    orthogonal lifted from D4
ρ182-22-2-22-220-2000022000-200000000    orthogonal lifted from D4
ρ1922-2-22-2-222000-2-200002000000000    orthogonal lifted from D4
ρ202-22-2-22-22020000-2-2000200000000    orthogonal lifted from D4
ρ212-2-22-2-22200220000-2-20000000000    orthogonal lifted from D4
ρ2222-2-22-2-22-2000220000-2000000000    orthogonal lifted from D4
ρ23222-2-2-22-20-20000-22000200000000    orthogonal lifted from D4
ρ2422-22-22-2-200-2200002-20000000000    orthogonal lifted from D4
ρ252-2-2-2222-220002-20000-2000000000    orthogonal lifted from D4
ρ2622-22-22-2-2002-20000-220000000000    orthogonal lifted from D4
ρ27222-2-2-22-20200002-2000-200000000    orthogonal lifted from D4
ρ282-2-22-2-22200-2-20000220000000000    orthogonal lifted from D4

Permutation representations of C2×C22≀C2
On 16 points - transitive group 16T105
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 3)(2 4)(5 12)(6 11)(7 14)(8 13)(9 15)(10 16)
(1 5)(2 6)(3 12)(4 11)(7 9)(8 10)(13 16)(14 15)
(1 5)(2 6)(3 12)(4 11)(7 15)(8 16)(9 14)(10 13)
(1 12)(2 11)(3 5)(4 6)(7 14)(8 13)(9 15)(10 16)
(1 10)(2 9)(3 8)(4 7)(5 13)(6 14)(11 15)(12 16)

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

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

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

G:=TransitiveGroup(16,105);

C2×C22≀C2 is a maximal subgroup of
C25.C22  C24.78D4  C24⋊D4  C23.203C24  C23.240C24  C23.257C24  C247D4  C23.304C24  C23.308C24  C248D4  C23.311C24  C24.95D4  C23.318C24  C23.324C24  C23.333C24  C23.372C24  C23.434C24  C23.439C24  C249D4  C2410D4  C24.97D4  C23.568C24  C23.569C24  C23.570C24  C23.578C24  C25⋊C22  C23.584C24  C23.585C24  C23.597C24  C2411D4  C23≀C2  C2413D4  C24.166D4  C2×D42  C22.73C25  C22.79C25  C42⋊C23
C2×C22≀C2 is a maximal quotient of
C23.288C24  C247D4  C23.304C24  C24.94D4  C24.243C23  C24.244C23  C23.308C24  C23.309C24  C248D4  C24.262C23  C24.263C23  C24.264C23  C23.333C24  C23.334C24  C23.335C24  C24.565C23  C249D4  C23.514C24  C24.360C23  C24.361C23  C23≀C2  C2413D4  C248Q8  C24.166D4  C24.103D4  C24.177D4  C24.178D4  C24.104D4  C24.105D4  C24.106D4  C4○D4⋊D4  D4.(C2×D4)  (C2×Q8)⋊16D4  Q8.(C2×D4)  (C2×D4)⋊21D4  (C2×Q8)⋊17D4  C42.313C23  M4(2)⋊C23  M4(2).C23  C42.12C23  C42.13C23  C23.7C24  C24⋊C23  C23.9C24  C23.10C24

Matrix representation of C2×C22≀C2 in GL6(ℤ)

100000
010000
00-1000
000-100
000010
000001
,
100000
0-10000
001000
000-100
0000-10
000001
,
-100000
0-10000
00-1000
000-100
0000-10
000001
,
-100000
0-10000
00-1000
000-100
0000-10
00000-1
,
100000
010000
001000
000100
0000-10
00000-1
,
0-10000
-100000
000100
001000
000001
000010

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,1],[-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,-1,0,0,0,0,0,0,1],[-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,-1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C22≀C2 in GAP, Magma, Sage, TeX

C_2\times C_2^2\wr C_2
% in TeX

G:=Group("C2xC2^2wrC2");
// GroupNames label

G:=SmallGroup(64,202);
// by ID

G=gap.SmallGroup(64,202);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650]);
// Polycyclic

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

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Character table of C2×C22≀C2 in TeX

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