p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.5C23, C4⋊Q8⋊3C22, (C2×D4).30D4, C8.2D4⋊1C2, (C2×Q8).30D4, C8⋊C4⋊28C22, C24⋊C22.C2, C2.27(D4⋊4D4), C4.4D4.9C22, C2.20(D4.9D4), C22.186C22≀C2, C42.C22⋊3C2, C42.28C22⋊26C2, (C2×C4).218(C2×D4), SmallGroup(128,391)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.5C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=dad=a-1, eae=a-1b2, cbc-1=ebe=b-1, dbd=a2b-1, dcd=ac, ece=bc, de=ed >
Subgroups: 336 in 114 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C24, C8⋊C4, C8⋊C4, D4⋊C4, Q8⋊C4, C22≀C2, C4.4D4, C4.4D4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C42.C22, C42.C22, C42.28C22, C8.2D4, C24⋊C22, C42.5C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, D4⋊4D4, D4.9D4, C42.5C23
Character table of C42.5C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ16 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | orthogonal lifted from D4⋊4D4 |
| ρ17 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | -2i | complex lifted from D4.9D4 |
| ρ18 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 2i | complex lifted from D4.9D4 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | complex lifted from D4.9D4 |
| ρ20 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from D4.9D4 |
►On 32 points
Generators in S32
(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 19 13 7)(2 20 14 8)(3 17 15 5)(4 18 16 6)(9 28 21 30)(10 25 22 31)(11 26 23 32)(12 27 24 29)
(1 27 3 25)(2 26 4 28)(5 22 7 24)(6 21 8 23)(9 20 11 18)(10 19 12 17)(13 29 15 31)(14 32 16 30)
(1 15)(2 14)(3 13)(4 16)(6 8)(9 10)(11 12)(18 20)(21 22)(23 24)(25 32)(26 31)(27 30)(28 29)
(2 16)(4 14)(5 17)(6 8)(7 19)(9 32)(10 25)(11 30)(12 27)(18 20)(21 26)(22 31)(23 28)(24 29)
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), (1,19,13,7)(2,20,14,8)(3,17,15,5)(4,18,16,6)(9,28,21,30)(10,25,22,31)(11,26,23,32)(12,27,24,29), (1,27,3,25)(2,26,4,28)(5,22,7,24)(6,21,8,23)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30), (1,15)(2,14)(3,13)(4,16)(6,8)(9,10)(11,12)(18,20)(21,22)(23,24)(25,32)(26,31)(27,30)(28,29), (2,16)(4,14)(5,17)(6,8)(7,19)(9,32)(10,25)(11,30)(12,27)(18,20)(21,26)(22,31)(23,28)(24,29)>;
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), (1,19,13,7)(2,20,14,8)(3,17,15,5)(4,18,16,6)(9,28,21,30)(10,25,22,31)(11,26,23,32)(12,27,24,29), (1,27,3,25)(2,26,4,28)(5,22,7,24)(6,21,8,23)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30), (1,15)(2,14)(3,13)(4,16)(6,8)(9,10)(11,12)(18,20)(21,22)(23,24)(25,32)(26,31)(27,30)(28,29), (2,16)(4,14)(5,17)(6,8)(7,19)(9,32)(10,25)(11,30)(12,27)(18,20)(21,26)(22,31)(23,28)(24,29) );
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)], [(1,19,13,7),(2,20,14,8),(3,17,15,5),(4,18,16,6),(9,28,21,30),(10,25,22,31),(11,26,23,32),(12,27,24,29)], [(1,27,3,25),(2,26,4,28),(5,22,7,24),(6,21,8,23),(9,20,11,18),(10,19,12,17),(13,29,15,31),(14,32,16,30)], [(1,15),(2,14),(3,13),(4,16),(6,8),(9,10),(11,12),(18,20),(21,22),(23,24),(25,32),(26,31),(27,30),(28,29)], [(2,16),(4,14),(5,17),(6,8),(7,19),(9,32),(10,25),(11,30),(12,27),(18,20),(21,26),(22,31),(23,28),(24,29)]])
Matrix representation of C42.5C23 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 8 | 9 | 8 | 8 | 0 | 0 | 0 | 0 |
| 9 | 9 | 8 | 9 | 0 | 0 | 0 | 0 |
| 8 | 8 | 8 | 9 | 0 | 0 | 0 | 0 |
| 8 | 9 | 9 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 | 2 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 15 | 15 |
| 0 | 0 | 0 | 0 | 2 | 15 | 2 | 15 |
| 0 | 0 | 0 | 0 | 15 | 15 | 15 | 15 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 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 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[8,9,8,8,0,0,0,0,9,9,8,9,0,0,0,0,8,8,8,9,0,0,0,0,8,9,9,9,0,0,0,0,0,0,0,0,15,2,2,15,0,0,0,0,2,2,15,15,0,0,0,0,2,15,2,15,0,0,0,0,15,15,15,15],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;
C42.5C23 in GAP, Magma, Sage, TeX
C_4^2._5C_2^3
% in TeX
G:=Group("C4^2.5C2^3"); // GroupNames label
G:=SmallGroup(128,391);
// by ID
G=gap.SmallGroup(128,391);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,1123,570,521,136,3924,1411,998,242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1,e*a*e=a^-1*b^2,c*b*c^-1=e*b*e=b^-1,d*b*d=a^2*b^-1,d*c*d=a*c,e*c*e=b*c,d*e=e*d>;
// generators/relations
Export