p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.53C23, C4.632+ 1+4, D42⋊9C2, C4⋊D8⋊38C2, C8⋊7D4⋊39C2, C8⋊2D4⋊25C2, C8⋊9D4⋊20C2, C4⋊C8⋊36C22, C4⋊C4.159D4, D4.Q8⋊37C2, D8⋊C4⋊23C2, C22⋊D8⋊31C2, (C2×D4).319D4, C2.49(D4○D8), (C4×D4)⋊26C22, (C2×D8)⋊10C22, C22⋊C4.52D4, C8⋊C4⋊25C22, C2.D8⋊13C22, C4.Q8⋊26C22, D4.26(C4○D4), C4⋊C4.237C23, C4⋊D4⋊17C22, C22⋊C8⋊32C22, (C2×C8).100C23, (C2×C4).510C24, (C22×C8)⋊32C22, C23.327(C2×D4), D4⋊C4⋊41C22, (C2×D4).236C23, C4⋊1D4.89C22, C2.146(D4⋊5D4), C42.C2⋊10C22, C42⋊C2⋊24C22, C23.46D4⋊17C2, C23.37D4⋊14C2, C23.19D4⋊35C2, C22.12(C8⋊C22), (C2×M4(2))⋊29C22, C22.770(C22×D4), C22.47C24⋊5C2, (C22×C4).1154C23, (C22×D4).413C22, C42.29C22⋊11C2, (C2×C4⋊C4)⋊60C22, C4.235(C2×C4○D4), (C2×C4).607(C2×D4), C2.77(C2×C8⋊C22), (C2×D4⋊C4)⋊32C2, SmallGroup(128,2050)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.53C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, ede-1=b2d >
Subgroups: 576 in 231 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C22×D4, C22×D4, C2×D4⋊C4, C23.37D4, C8⋊9D4, D8⋊C4, C22⋊D8, C4⋊D8, C8⋊7D4, C8⋊2D4, D4.Q8, C23.46D4, C23.19D4, C42.29C22, D42, C22.47C24, C42.53C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C8⋊C22, D4○D8, C42.53C23
Character table of C42.53C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -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 | 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 | -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 | -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 | 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 | 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 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 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 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -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 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -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 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -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 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| ρ15 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 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 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -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 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ27 | 4 | -4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
►On 32 points
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 28 24 19)(2 25 21 20)(3 26 22 17)(4 27 23 18)(5 12 15 31)(6 9 16 32)(7 10 13 29)(8 11 14 30)
(1 29 3 31)(2 9 4 11)(5 19 7 17)(6 27 8 25)(10 22 12 24)(13 26 15 28)(14 20 16 18)(21 32 23 30)
(1 3)(2 23)(4 21)(5 10)(6 30)(7 12)(8 32)(9 14)(11 16)(13 31)(15 29)(17 28)(18 20)(19 26)(22 24)(25 27)
(1 23 24 4)(2 3 21 22)(5 14 15 8)(6 7 16 13)(9 10 32 29)(11 12 30 31)(17 25 26 20)(18 19 27 28)
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,29,3,31)(2,9,4,11)(5,19,7,17)(6,27,8,25)(10,22,12,24)(13,26,15,28)(14,20,16,18)(21,32,23,30), (1,3)(2,23)(4,21)(5,10)(6,30)(7,12)(8,32)(9,14)(11,16)(13,31)(15,29)(17,28)(18,20)(19,26)(22,24)(25,27), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,28,24,19)(2,25,21,20)(3,26,22,17)(4,27,23,18)(5,12,15,31)(6,9,16,32)(7,10,13,29)(8,11,14,30), (1,29,3,31)(2,9,4,11)(5,19,7,17)(6,27,8,25)(10,22,12,24)(13,26,15,28)(14,20,16,18)(21,32,23,30), (1,3)(2,23)(4,21)(5,10)(6,30)(7,12)(8,32)(9,14)(11,16)(13,31)(15,29)(17,28)(18,20)(19,26)(22,24)(25,27), (1,23,24,4)(2,3,21,22)(5,14,15,8)(6,7,16,13)(9,10,32,29)(11,12,30,31)(17,25,26,20)(18,19,27,28) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,28,24,19),(2,25,21,20),(3,26,22,17),(4,27,23,18),(5,12,15,31),(6,9,16,32),(7,10,13,29),(8,11,14,30)], [(1,29,3,31),(2,9,4,11),(5,19,7,17),(6,27,8,25),(10,22,12,24),(13,26,15,28),(14,20,16,18),(21,32,23,30)], [(1,3),(2,23),(4,21),(5,10),(6,30),(7,12),(8,32),(9,14),(11,16),(13,31),(15,29),(17,28),(18,20),(19,26),(22,24),(25,27)], [(1,23,24,4),(2,3,21,22),(5,14,15,8),(6,7,16,13),(9,10,32,29),(11,12,30,31),(17,25,26,20),(18,19,27,28)]])
Matrix representation of C42.53C23 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 16 | 16 | 16 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 14 | 11 | 0 | 0 |
| 0 | 0 | 11 | 14 | 14 | 3 |
| 0 | 0 | 3 | 3 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 | 16 | 0 |
G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,1,1,16,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,6,14,11,3,0,0,6,11,14,3,0,0,0,0,14,3,0,0,0,0,3,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,16,0,1,16,0,0,0,0,0,16,0,0,15,1,1,16,0,0,0,1,0,0] >;
C42.53C23 in GAP, Magma, Sage, TeX
C_4^2._{53}C_2^3 % in TeX
G:=Group("C4^2.53C2^3"); // GroupNames label
G:=SmallGroup(128,2050);
// by ID
G=gap.SmallGroup(128,2050);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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