p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊1C4, C4.9C42, C23.8D4, (C2×C8)⋊1C4, (C2×C4).8D4, C4.1(C4⋊C4), (C2×C4).1Q8, C22.1(C4⋊C4), C4.17(C22⋊C4), C42⋊C2.1C2, (C2×M4(2)).4C2, C22.6(C22⋊C4), (C22×C4).18C22, C2.2(C2.C42), (C2×C4).61(C2×C4), SmallGroup(64,18)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.9C42
G = < a,b,c | a4=b4=c4=1, cbc-1=ab=ba, ac=ca >
Character table of C4.9C42
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 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 | i | -i | -i | i | i | i | -i | -i | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | -1 | 1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | -1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | -i | i | -i | -i | i | i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | i | -i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -i | -i | i | i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | -1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | i | -i | -i | -i | i | i | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | i | i | -i | i | i | -i | -i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | i | i | -i | -i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | i | -i | i | i | -i | -i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -i | i | -i | -1 | 1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T74
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5 11 16)(2 6 12 13)(3 7 9 14)(4 8 10 15)
(1 11 3 9)(2 12 4 10)(5 15)(6 16)(7 13)(8 14)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5,11,16)(2,6,12,13)(3,7,9,14)(4,8,10,15), (1,11,3,9)(2,12,4,10)(5,15)(6,16)(7,13)(8,14)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5,11,16)(2,6,12,13)(3,7,9,14)(4,8,10,15), (1,11,3,9)(2,12,4,10)(5,15)(6,16)(7,13)(8,14) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5,11,16),(2,6,12,13),(3,7,9,14),(4,8,10,15)], [(1,11,3,9),(2,12,4,10),(5,15),(6,16),(7,13),(8,14)]])
G:=TransitiveGroup(16,74);
►On 16 points - transitive group 16T123
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11)(2 12)(3 9)(4 10)(5 15 7 13)(6 16 8 14)
(1 7)(2 8)(3 5)(4 6)(9 14 11 16)(10 15 12 13)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11)(2,12)(3,9)(4,10)(5,15,7,13)(6,16,8,14), (1,7)(2,8)(3,5)(4,6)(9,14,11,16)(10,15,12,13)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,11)(2,12)(3,9)(4,10)(5,15,7,13)(6,16,8,14), (1,7)(2,8)(3,5)(4,6)(9,14,11,16)(10,15,12,13) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,11),(2,12),(3,9),(4,10),(5,15,7,13),(6,16,8,14)], [(1,7),(2,8),(3,5),(4,6),(9,14,11,16),(10,15,12,13)]])
G:=TransitiveGroup(16,123);
C4.9C42 is a maximal subgroup of
C23.D8 C23.2D8 (C2×D4).24Q8 (C2×C42)⋊C4 C8.(C4⋊C4) C8⋊C4⋊17C4 C42.5D4 C42.6D4 C4.(C4×D4) (C2×C8)⋊4D4 C42⋊D4 C42.7D4 C42⋊2D4 C42.8D4 C22⋊C4.7D4 C42.9D4 (C2×C8).D4 (C2×C8).6D4 C42.10D4 C42.32Q8
C4p.C42: C8.16C42 C42⋊3Dic3 C12.20C42 C42⋊1Dic5 C23.9D20 C42⋊3F5 C20.24C42 C42⋊Dic7 ...
C4.9C42 is a maximal quotient of
C42.20D4 C24.46D4 C42.4Q8 C42.23D4 C42.25D4 C20.24C42
C23.D4p: C23.8D8 C12.20C42 C23.9D20 C23.9D28 ...
(C4×C4p)⋊C4: C42⋊1C8 C42.5Q8 C42.6Q8 C42⋊3Dic3 C42⋊1Dic5 C42⋊3F5 C42⋊Dic7 ...
Matrix representation of C4.9C42 ►in GL4(𝔽5) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,4,0,0,0,0,2,0,0,0,0,1,2,0,0,0],[3,0,0,0,0,4,0,0,0,0,2,0,0,0,0,1] >;
C4.9C42 in GAP, Magma, Sage, TeX
C_4._9C_4^2
% in TeX
G:=Group("C4.9C4^2"); // GroupNames label
G:=SmallGroup(64,18);
// by ID
G=gap.SmallGroup(64,18);
# by ID
G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,158,489,1444]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^4=1,c*b*c^-1=a*b=b*a,a*c=c*a>;
// generators/relations
Export
Subgroup lattice of C4.9C42 in TeX
Character table of C4.9C42 in TeX