p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.2C23, (C2×D4)⋊6C4, (C2×Q8)⋊4C4, C4○(C23⋊C4), C23⋊C4⋊5C2, (C22×C4)⋊4C4, (C2×C4).119D4, C23.3(C2×C4), C22.9(C2×D4), C42⋊C2⋊1C2, 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
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, C4, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C23⋊C4, C42⋊C2, C2×C4○D4, C23.C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C23.C23
Character table of C23.C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | -1 | -i | i | -i | i | -i | i | -1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | i | i | -1 | i | -i | -i | -i | -i | i | 1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -1 | i | -i | i | -i | i | -i | -1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | -i | -1 | -i | i | i | i | i | -i | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | 1 | -i | -i | -i | i | i | -i | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -i | i | 1 | i | i | -i | -i | i | -i | 1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | 1 | i | i | i | -i | -i | i | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | i | -i | 1 | -i | -i | i | i | -i | i | 1 | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T101
(1 5)(4 8)(10 12)(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 13)(2 14)(3 11)(4 10)(5 15)(6 16)(7 9)(8 12)
(1 4 5 8)(2 3 6 7)(9 14 11 16)(10 15 12 13)
G:=sub<Sym(16)| (1,5)(4,8)(10,12)(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), (1,13)(2,14)(3,11)(4,10)(5,15)(6,16)(7,9)(8,12), (1,4,5,8)(2,3,6,7)(9,14,11,16)(10,15,12,13)>;
G:=Group( (1,5)(4,8)(10,12)(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), (1,13)(2,14)(3,11)(4,10)(5,15)(6,16)(7,9)(8,12), (1,4,5,8)(2,3,6,7)(9,14,11,16)(10,15,12,13) );
G=PermutationGroup([[(1,5),(4,8),(10,12),(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)], [(1,13),(2,14),(3,11),(4,10),(5,15),(6,16),(7,9),(8,12)], [(1,4,5,8),(2,3,6,7),(9,14,11,16),(10,15,12,13)]])
G:=TransitiveGroup(16,101);
►On 16 points - transitive group 16T112
(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
(1 16)(2 4)(3 14)(5 7)(6 11)(8 9)(10 12)(13 15)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 14)(2 15)(3 16)(4 13)(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 10)(2 6)(3 5)(4 11)(7 14)(8 13)(9 15)(12 16)
(1 9 14 6)(2 10 15 7)(3 11 16 8)(4 12 13 5)
G:=sub<Sym(16)| (1,16)(2,4)(3,14)(5,7)(6,11)(8,9)(10,12)(13,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,14)(2,15)(3,16)(4,13)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,6)(3,5)(4,11)(7,14)(8,13)(9,15)(12,16), (1,9,14,6)(2,10,15,7)(3,11,16,8)(4,12,13,5)>;
G:=Group( (1,16)(2,4)(3,14)(5,7)(6,11)(8,9)(10,12)(13,15), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,14)(2,15)(3,16)(4,13)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,6)(3,5)(4,11)(7,14)(8,13)(9,15)(12,16), (1,9,14,6)(2,10,15,7)(3,11,16,8)(4,12,13,5) );
G=PermutationGroup([[(1,16),(2,4),(3,14),(5,7),(6,11),(8,9),(10,12),(13,15)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,14),(2,15),(3,16),(4,13),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10),(2,6),(3,5),(4,11),(7,14),(8,13),(9,15),(12,16)], [(1,9,14,6),(2,10,15,7),(3,11,16,8),(4,12,13,5)]])
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⋊F5⋊5C2 (D4×C14)⋊10C4 ...
(C2×D4).D2p: 2+ 1+4⋊4C4 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⋊F5⋊5C2 (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⋊C4⋊5S3 (C6×D4)⋊10C4 C23⋊C4⋊5D5 (D4×C10)⋊22C4 C23⋊C4⋊5D7 (D4×C14)⋊10C4 ...
Matrix representation of C23.C23 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 1 | 1 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 1 | 4 | 3 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 1 | 1 | 4 | 3 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 3 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 3 | 3 | 2 | 4 |
| 2 | 0 | 3 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
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
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