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G = C23.C23order 64 = 26

2nd non-split extension by C23 of C23 acting via C23/C2=C22

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

Aliases: C23.2C23, (C2×D4)⋊6C4, (C2×Q8)⋊4C4, C4(C23⋊C4), C23⋊C45C2, (C22×C4)⋊4C4, (C2×C4).119D4, C23.3(C2×C4), C22.9(C2×D4), C42⋊C21C2, C4.9(C22⋊C4), (C2×D4).43C22, C22.7(C22×C4), C22⋊C4.10C22, (C22×C4).30C22, (C2×C4).5(C2×C4), (C2×C4○D4).2C2, C2.13(C2×C22⋊C4), SmallGroup(64,91)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.C23
C1C2C22C23C22×C4C2×C4○D4 — C23.C23
C1C2C22 — C23.C23
C1C4C22×C4 — C23.C23
C1C2C23 — C23.C23

Generators and relations for C23.C23
 G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=b, f2=c, ab=ba, dad-1=ac=ca, ae=ea, af=fa, ebe=bc=cb, bd=db, bf=fb, cd=dc, ce=ec, cf=fc, ede=acd, df=fd, ef=fe >

Subgroups: 137 in 79 conjugacy classes, 39 normal (13 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C2×C4 [×2], C2×C4 [×6], C2×C4 [×8], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C23⋊C4 [×4], C42⋊C2 [×2], C2×C4○D4, C23.C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C23.C23

Character table of C23.C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 1122244112224444444444
ρ11111111111111111111111    trivial
ρ211111-11-1-1-1-1-1-111-1-11-111-1    linear of order 2
ρ3111111-1-1-1-1-1-1-11-11-111-1-11    linear of order 2
ρ411111-1-11111111-1-111-1-1-1-1    linear of order 2
ρ5111111111111-1-11-1-1-1-1-1-11    linear of order 2
ρ611111-11-1-1-1-1-11-1111-11-1-1-1    linear of order 2
ρ7111111-1-1-1-1-1-11-1-1-11-1-1111    linear of order 2
ρ811111-1-111111-1-1-11-1-1111-1    linear of order 2
ρ911-11-111-1-111-1-ii-1-ii-ii-ii-1    linear of order 4
ρ1011-11-1-1111-1-11ii-1i-i-i-i-ii1    linear of order 4
ρ1111-11-111-1-111-1i-i-1i-ii-ii-i-1    linear of order 4
ρ1211-11-1-1111-1-11-i-i-1-iiiii-i1    linear of order 4
ρ1311-11-11-111-1-11ii1-i-i-iii-i-1    linear of order 4
ρ1411-11-1-1-1-1-111-1-ii1ii-i-ii-i1    linear of order 4
ρ1511-11-11-111-1-11-i-i1iii-i-ii-1    linear of order 4
ρ1611-11-1-1-1-1-111-1i-i1-i-iii-ii1    linear of order 4
ρ1722-2-2200222-2-20000000000    orthogonal lifted from D4
ρ18222-2-200-2-22-220000000000    orthogonal lifted from D4
ρ19222-2-20022-22-20000000000    orthogonal lifted from D4
ρ2022-2-2200-2-2-2220000000000    orthogonal lifted from D4
ρ214-400000-4i4i0000000000000    complex faithful
ρ224-4000004i-4i0000000000000    complex faithful

Permutation representations of C23.C23
On 16 points - transitive group 16T101
Generators in S16
(1 5)(4 8)(9 11)(13 15)
(9 11)(10 12)(13 15)(14 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 15)(2 16)(3 12)(4 11)(5 13)(6 14)(7 10)(8 9)
(1 4 5 8)(2 3 6 7)(9 15 11 13)(10 16 12 14)

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

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

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

G:=TransitiveGroup(16,101);

On 16 points - transitive group 16T112
Generators in S16
(2 16)(4 14)(6 9)(8 11)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(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)
(2 16)(3 13)(5 12)(8 11)
(1 10 15 7)(2 11 16 8)(3 12 13 5)(4 9 14 6)

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

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

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

G:=TransitiveGroup(16,112);

On 16 points - transitive group 16T120
Generators in S16
(1 14)(2 4)(3 16)(5 7)(6 9)(8 11)(10 12)(13 15)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 16)(2 13)(3 14)(4 15)(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 10)(2 8)(3 7)(4 11)(5 16)(6 15)(9 13)(12 14)
(1 9 16 8)(2 10 13 5)(3 11 14 6)(4 12 15 7)

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

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

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

G:=TransitiveGroup(16,120);

C23.C23 is a maximal subgroup of
(C2×D4).Q8  C42.6D4  M4(2)⋊5D4  C42.8D4  C23.4C24  C23.7C24  C23.9C24  (C2×D4)⋊8F5
 (C2×D4p)⋊C4: C42.5D4  M4(2).30D4  (C2×D12)⋊13C4  (C2×D20)⋊25C4  (C2×Q8)⋊7F5  (C2×D28)⋊13C4 ...
 (C22×C4p)⋊C4: C23.2C42  (C22×C8)⋊C4  C23.5C42  (C2×D4).24Q8  (C6×D4)⋊10C4  (D4×C10)⋊22C4  C23⋊F55C2  (D4×C14)⋊10C4 ...
 (C2×D4).D2p: 2+ 1+44C4  C4○D4.D4  (C22×Q8)⋊C4  (C2×C42)⋊C4  M4(2)⋊19D4  C42.426D4  C4.(C4×D4)  (C2×C8)⋊4D4 ...
C23.C23 is a maximal quotient of
C42.371D4  C42.393D4  C42.394D4  C42.42D4  C42.43D4  C42.44D4  C42.395D4  C42.396D4  C42.372D4  C42.375D4  C42.404D4  C42.55D4  C42.56D4  C42.57D4  C42.58D4  C42.59D4  C42.60D4  C42.62D4  C42.63D4  C24.162C23  C4×C23⋊C4  C24.169C23  C24.174C23  C24.175C23  C24.176C23  C23⋊F55C2  (C2×D4)⋊8F5  (C2×Q8)⋊7F5
 (C2×C4).D4p: C42.403D4  C42.61D4  (C2×D12)⋊13C4  (C2×D20)⋊25C4  (C2×D28)⋊13C4 ...
 (C2×D4).D2p: C24.165C23  C24.167C23  C23⋊C45S3  (C6×D4)⋊10C4  C23⋊C45D5  (D4×C10)⋊22C4  C23⋊C45D7  (D4×C14)⋊10C4 ...

Matrix representation of C23.C23 in GL4(𝔽5) generated by

1000
0100
0040
1104
,
0100
1000
1143
0001
,
4000
0400
0040
0004
,
0010
1143
0100
0004
,
0300
2000
3324
2033
,
3000
0300
0030
0003
G:=sub<GL(4,GF(5))| [1,0,0,1,0,1,0,1,0,0,4,0,0,0,0,4],[0,1,1,0,1,0,1,0,0,0,4,0,0,0,3,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,0,1,1,0,1,4,0,0,0,3,0,4],[0,2,3,2,3,0,3,0,0,0,2,3,0,0,4,3],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;

C23.C23 in GAP, Magma, Sage, TeX

C_2^3.C_2^3
% in TeX

G:=Group("C2^3.C2^3");
// GroupNames label

G:=SmallGroup(64,91);
// by ID

G=gap.SmallGroup(64,91);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,158,963,730]);
// Polycyclic

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

Export

Character table of C23.C23 in TeX

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