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G = C22.54C24order 64 = 26

40th central stem extension by C22 of C24

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

Aliases: C4211C22, C23.23C23, C22.54C24, C24.21C22, C2.212+ 1+4, C41D49C2, C4⋊C46C22, C22≀C27C2, C4⋊D417C2, (C2×D4)⋊7C22, C422C28C2, C22⋊C49C22, (C2×C4).36C23, (C22×C4)⋊12C22, C22.D413C2, SmallGroup(64,241)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.54C24
C1C2C22C23C24C22≀C2 — C22.54C24
C1C22 — C22.54C24
C1C22 — C22.54C24
C1C22 — C22.54C24

Generators and relations for C22.54C24
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f2=1, ab=ba, dcd=ac=ca, fdf=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, fcf=abc, ede=abd, ef=fe >

Subgroups: 237 in 126 conjugacy classes, 71 normal (7 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C2×C4 [×9], C2×C4 [×3], D4 [×12], C23, C23 [×5], C23 [×3], C42, C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×12], C24, C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C422C2 [×2], C41D4, C22.54C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2+ 1+4 [×3], C22.54C24

Character table of C22.54C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I
 size 1111444444444444444
ρ11111111111111111111    trivial
ρ21111-11-111111-1-1-1-1-1-11    linear of order 2
ρ31111-11-1-11-11-1-11-1111-1    linear of order 2
ρ41111111-11-11-11-11-1-1-1-1    linear of order 2
ρ5111111-1-1-11-111-1-111-1-1    linear of order 2
ρ61111-111-1-11-11-111-1-11-1    linear of order 2
ρ71111-1111-1-1-1-1-1-1111-11    linear of order 2
ρ8111111-11-1-1-1-111-1-1-111    linear of order 2
ρ91111-1-1-11-1-111111-11-1-1    linear of order 2
ρ1011111-111-1-111-1-1-11-11-1    linear of order 2
ρ1111111-11-1-111-1-11-1-11-11    linear of order 2
ρ121111-1-1-1-1-111-11-111-111    linear of order 2
ρ131111-1-11-11-1-111-1-1-1111    linear of order 2
ρ1411111-1-1-11-1-11-1111-1-11    linear of order 2
ρ1511111-1-1111-1-1-1-11-111-1    linear of order 2
ρ161111-1-11111-1-111-11-1-1-1    linear of order 2
ρ174-4-44000000000000000    orthogonal lifted from 2+ 1+4
ρ184-44-4000000000000000    orthogonal lifted from 2+ 1+4
ρ1944-4-4000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of C22.54C24
On 16 points - transitive group 16T83
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 5)(2 6)(3 15)(4 16)(7 10)(8 9)(11 14)(12 13)
(1 12)(2 11)(3 10)(4 9)(5 13)(6 14)(7 15)(8 16)
(1 8)(2 7)(3 13)(4 14)(5 9)(6 10)(11 16)(12 15)
(1 6)(2 5)(3 15)(4 16)(11 12)(13 14)
(3 15)(4 16)(7 8)(9 10)(11 13)(12 14)

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

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

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

G:=TransitiveGroup(16,83);

C22.54C24 is a maximal subgroup of
C42⋊C23  C22.122C25  C22.123C25  C22.149C25  C22.155C25  C24.6A4
 C42⋊D2p: C426D4  C4227D6  C4230D6  C4225D10  C4228D10  C4225D14  C4228D14 ...
 C2p.2+ 1+4: C22.118C25  C22.126C25  C22.128C25  C22.129C25  C22.131C25  C22.132C25  C22.135C25  C22.140C25 ...
C22.54C24 is a maximal quotient of
C23.257C24  C24.225C23  C23.259C24  C23.262C24  C24.230C23  C23.568C24  C23.569C24  C24.384C23  C23.578C24  C23.585C24  C24.395C23  C23.591C24  C23.593C24  C24.406C23  C23.603C24  C23.635C24  C23.637C24  C24.428C23  C24.432C23  C24.434C23  C23.649C24  C23.652C24  C23.656C24  C24.438C23  C23.660C24  C23.678C24  C24.448C23  C24.450C23  C24.454C23  C23.692C24  C23.695C24  C23.696C24  C23.697C24  C23.701C24  C23.703C24  C23.707C24  C2411D4  C24.459C23  C23.715C24  C23.724C24  C23.725C24  C23.726C24  C23.727C24  C23.728C24  C23.729C24  C23.730C24  C23.734C24  C23.737C24  C23.738C24  C23.741C24  C24.15Q8  C4212Q8
 C42⋊D2p: C4233D4  C4235D4  C4227D6  C4230D6  C4225D10  C4228D10  C4225D14  C4228D14 ...
 C4⋊C4⋊D2p: C23.573C24  C23.605C24  C24.413C23  C6.482+ 1+4  C6.682+ 1+4  C10.482+ 1+4  C10.682+ 1+4  C14.482+ 1+4 ...
 (C2×D4).D2p: C23.597C24  C24.411C23  C24.47D6  C24.36D10  C24.36D14 ...

Matrix representation of C22.54C24 in GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
0-1-200000
-100-20000
00010000
00100000
00000-1-10
0000-100-1
00000001
00000010
,
01000000
10000000
00010000
00100000
00000100
00001000
0000-200-1
00000-2-10
,
-10000000
01000000
00100000
000-10000
00001000
00000-100
00000210
0000-200-1
,
10000000
01000000
0-1-100000
-100-10000
00001000
00000-100
000000-10
00000001

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,0,0,1,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,-1,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,-2,0,0,0,0,0,1,0,0,-2,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-2,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1] >;

C22.54C24 in GAP, Magma, Sage, TeX

C_2^2._{54}C_2^4
% in TeX

G:=Group("C2^2.54C2^4");
// GroupNames label

G:=SmallGroup(64,241);
// by ID

G=gap.SmallGroup(64,241);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650,476,1347,297]);
// 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,d*c*d=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f=a*b*c,e*d*e=a*b*d,e*f=f*e>;
// generators/relations

Export

Character table of C22.54C24 in TeX

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