p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.410C23, C4.1162+ 1+4, C8⋊3D4⋊15C2, C4⋊D8⋊28C2, C4⋊C8⋊22C22, C4⋊C4.133D4, (C4×C8)⋊17C22, C4⋊Q8⋊13C22, D4⋊D4⋊30C2, C22⋊D8⋊24C2, C4⋊SD16⋊12C2, C2.31(D4○D8), C4.4D8⋊18C2, C8.12D4⋊5C2, (C4×D4)⋊16C22, (C2×D8)⋊26C22, C22⋊C4.25D4, (C2×Q16)⋊6C22, (C4×Q8)⋊16C22, C8⋊C4⋊13C22, C22⋊SD16⋊13C2, D4.7D4⋊30C2, D4.2D4⋊27C2, D4⋊C4⋊4C22, C4⋊C4.163C23, (C2×C8).163C23, (C2×C4).422C24, Q8.D4⋊27C2, C23.294(C2×D4), C2.47(D4○SD16), Q8⋊C4⋊35C22, (C2×SD16)⋊45C22, (C2×D4).171C23, C4⋊D4.45C22, C4.4D4⋊10C22, C4⋊1D4.69C22, C22⋊C8.57C22, (C2×Q8).159C23, C22.29C24⋊17C2, C22⋊Q8.45C22, (C22×C4).310C23, C22.682(C22×D4), C42.28C22⋊5C2, C42.7C22⋊14C2, C22.36C24⋊7C2, (C22×D4).394C22, C42⋊C2.161C22, C2.93(C22.29C24), (C2×C4).551(C2×D4), (C2×C4○D4).181C22, SmallGroup(128,1956)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.410C23
G = < a,b,c,d,e | a4=b4=c2=d2=e2=1, ab=ba, cac=dad=a-1b2, eae=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece=a2b2c, de=ed >
Subgroups: 500 in 204 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C22×D4, C2×C4○D4, C42.7C22, C22⋊D8, D4⋊D4, C22⋊SD16, D4.7D4, C4⋊D8, C4⋊SD16, D4.2D4, Q8.D4, C4.4D8, C42.28C22, C8.12D4, C8⋊3D4, C22.29C24, C22.36C24, C42.410C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, D4○D8, D4○SD16, C42.410C23
Character table of C42.410C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -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 | 0 | 0 | -2 | -2 | 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 | 0 | 0 | -2 | -2 | 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 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 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 |
| ρ22 | 4 | -4 | 4 | -4 | 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 |
| ρ23 | 4 | -4 | -4 | 4 | 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 |
| ρ24 | 4 | -4 | -4 | 4 | 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 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
►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 24 26 19)(2 21 27 20)(3 22 28 17)(4 23 25 18)(5 12 31 15)(6 9 32 16)(7 10 29 13)(8 11 30 14)
(1 5)(2 30)(3 7)(4 32)(6 25)(8 27)(9 23)(10 17)(11 21)(12 19)(13 22)(14 20)(15 24)(16 18)(26 31)(28 29)
(1 19)(2 23)(3 17)(4 21)(6 30)(8 32)(9 11)(10 13)(12 15)(14 16)(18 27)(20 25)(22 28)(24 26)
(2 27)(4 25)(5 29)(6 8)(7 31)(9 11)(10 15)(12 13)(14 16)(18 23)(20 21)(30 32)
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), (1,24,26,19)(2,21,27,20)(3,22,28,17)(4,23,25,18)(5,12,31,15)(6,9,32,16)(7,10,29,13)(8,11,30,14), (1,5)(2,30)(3,7)(4,32)(6,25)(8,27)(9,23)(10,17)(11,21)(12,19)(13,22)(14,20)(15,24)(16,18)(26,31)(28,29), (1,19)(2,23)(3,17)(4,21)(6,30)(8,32)(9,11)(10,13)(12,15)(14,16)(18,27)(20,25)(22,28)(24,26), (2,27)(4,25)(5,29)(6,8)(7,31)(9,11)(10,15)(12,13)(14,16)(18,23)(20,21)(30,32)>;
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), (1,24,26,19)(2,21,27,20)(3,22,28,17)(4,23,25,18)(5,12,31,15)(6,9,32,16)(7,10,29,13)(8,11,30,14), (1,5)(2,30)(3,7)(4,32)(6,25)(8,27)(9,23)(10,17)(11,21)(12,19)(13,22)(14,20)(15,24)(16,18)(26,31)(28,29), (1,19)(2,23)(3,17)(4,21)(6,30)(8,32)(9,11)(10,13)(12,15)(14,16)(18,27)(20,25)(22,28)(24,26), (2,27)(4,25)(5,29)(6,8)(7,31)(9,11)(10,15)(12,13)(14,16)(18,23)(20,21)(30,32) );
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)], [(1,24,26,19),(2,21,27,20),(3,22,28,17),(4,23,25,18),(5,12,31,15),(6,9,32,16),(7,10,29,13),(8,11,30,14)], [(1,5),(2,30),(3,7),(4,32),(6,25),(8,27),(9,23),(10,17),(11,21),(12,19),(13,22),(14,20),(15,24),(16,18),(26,31),(28,29)], [(1,19),(2,23),(3,17),(4,21),(6,30),(8,32),(9,11),(10,13),(12,15),(14,16),(18,27),(20,25),(22,28),(24,26)], [(2,27),(4,25),(5,29),(6,8),(7,31),(9,11),(10,15),(12,13),(14,16),(18,23),(20,21),(30,32)]])
Matrix representation of C42.410C23 ►in GL8(𝔽17)
| 1 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 1 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 16 | 1 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 5 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 5 | 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 3 | 3 |
| 0 | 0 | 0 | 0 | 14 | 3 | 3 | 14 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,16,0,1,0,0,0,0,2,1,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,5,0,12,0,0,0,0,0,0,5,5,0,0,0,0,7,7,12,12,0,0,0,0,10,0,5,5,0,0,0,0,0,0,0,0,6,3,0,14,0,0,0,0,11,11,3,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,3,14],[1,0,1,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,1,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
C42.410C23 in GAP, Magma, Sage, TeX
C_4^2._{410}C_2^3 % in TeX
G:=Group("C4^2.410C2^3"); // GroupNames label
G:=SmallGroup(128,1956);
// by ID
G=gap.SmallGroup(128,1956);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^2=1,a*b=b*a,c*a*c=d*a*d=a^-1*b^2,e*a*e=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*b^2*c,d*e=e*d>;
// generators/relations
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