p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.49C23, C4.592+ 1+4, C8⋊9D4⋊16C2, C8⋊D4⋊38C2, C4⋊C8⋊35C22, C4⋊C4.157D4, D4⋊3Q8⋊3C2, C4⋊Q8⋊24C22, D4⋊Q8⋊34C2, (C2×D4).317D4, D4⋊5D4.4C2, C2.48(D4○D8), (C2×C8).98C23, (C2×Q16)⋊9C22, (C4×Q8)⋊28C22, C8⋊C4⋊24C22, C2.D8⋊37C22, D4.24(C4○D4), C22⋊SD16⋊22C2, D4.7D4⋊44C2, C8.18D4⋊39C2, C4⋊C4.235C23, C22⋊C8⋊31C22, (C2×C4).506C24, Q8.D4⋊42C2, C22⋊C4.167D4, C23.475(C2×D4), C22⋊Q8⋊19C22, SD16⋊C4⋊35C2, Q8⋊C4⋊44C22, (C4×D4).159C22, (C2×D4).423C23, C22.D8⋊28C2, C4⋊D4.84C22, (C2×Q8).219C23, C2.142(D4⋊5D4), D4⋊C4.72C22, C23.36D4⋊17C2, C23.48D4⋊26C2, (C2×M4(2))⋊28C22, (C22×C8).309C22, (C2×SD16).56C22, C4.4D4.66C22, C22.766(C22×D4), C22.5(C8.C22), (C22×C4).1150C23, (C22×D4).412C22, C42.28C22⋊16C2, (C2×C4⋊C4)⋊59C22, C4.231(C2×C4○D4), (C2×C4).603(C2×D4), (C2×D4⋊C4)⋊31C2, C2.75(C2×C8.C22), (C2×C4○D4).210C22, SmallGroup(128,2046)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.49C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2b2, ab=ba, cac-1=eae=a-1, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >
Subgroups: 448 in 205 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C22×D4, C2×C4○D4, C2×D4⋊C4, C23.36D4, C8⋊9D4, SD16⋊C4, C22⋊SD16, D4.7D4, Q8.D4, C8.18D4, C8⋊D4, D4⋊Q8, C22.D8, C23.48D4, C42.28C22, D4⋊5D4, D4⋊3Q8, C42.49C23
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, C42.49C23
Character table of C42.49C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 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 | -2 | -2 | 0 | 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 | 0 | 0 | -2 | 0 | -2 | -2 | 0 | 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 | 0 | 0 | -2 | 0 | -2 | -2 | 0 | -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 | 0 | 0 | 2 | 0 | -2 | -2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 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 | -2 | 2 | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 2 | -2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 2 | -2i | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 2 | 2i | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 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 | 0 | 2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | -2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ28 | 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 | symplectic lifted from C8.C22, Schur index 2 |
| ρ29 | 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 | symplectic lifted from C8.C22, Schur index 2 |
►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 19 23 25)(2 20 24 26)(3 17 21 27)(4 18 22 28)(5 15 9 32)(6 16 10 29)(7 13 11 30)(8 14 12 31)
(1 6 21 12)(2 5 22 11)(3 8 23 10)(4 7 24 9)(13 20 32 28)(14 19 29 27)(15 18 30 26)(16 17 31 25)
(1 21)(2 4)(3 23)(5 30)(6 14)(7 32)(8 16)(9 13)(10 31)(11 15)(12 29)(17 19)(18 26)(20 28)(22 24)(25 27)
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)
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), (1,19,23,25)(2,20,24,26)(3,17,21,27)(4,18,22,28)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,6,21,12)(2,5,22,11)(3,8,23,10)(4,7,24,9)(13,20,32,28)(14,19,29,27)(15,18,30,26)(16,17,31,25), (1,21)(2,4)(3,23)(5,30)(6,14)(7,32)(8,16)(9,13)(10,31)(11,15)(12,29)(17,19)(18,26)(20,28)(22,24)(25,27), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)>;
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), (1,19,23,25)(2,20,24,26)(3,17,21,27)(4,18,22,28)(5,15,9,32)(6,16,10,29)(7,13,11,30)(8,14,12,31), (1,6,21,12)(2,5,22,11)(3,8,23,10)(4,7,24,9)(13,20,32,28)(14,19,29,27)(15,18,30,26)(16,17,31,25), (1,21)(2,4)(3,23)(5,30)(6,14)(7,32)(8,16)(9,13)(10,31)(11,15)(12,29)(17,19)(18,26)(20,28)(22,24)(25,27), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31) );
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)], [(1,19,23,25),(2,20,24,26),(3,17,21,27),(4,18,22,28),(5,15,9,32),(6,16,10,29),(7,13,11,30),(8,14,12,31)], [(1,6,21,12),(2,5,22,11),(3,8,23,10),(4,7,24,9),(13,20,32,28),(14,19,29,27),(15,18,30,26),(16,17,31,25)], [(1,21),(2,4),(3,23),(5,30),(6,14),(7,32),(8,16),(9,13),(10,31),(11,15),(12,29),(17,19),(18,26),(20,28),(22,24),(25,27)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31)]])
Matrix representation of C42.49C23 ►in GL6(𝔽17)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 1 | 1 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 1 | 15 |
| 0 | 0 | 16 | 0 | 1 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 6 |
| 0 | 0 | 3 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 | 16 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 1 | 1 | 15 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,16,16,16,0,0,0,1,0,1,0,0,1,1,0,0,0,0,0,15,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,16,0,0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,15,16],[13,4,0,0,0,0,0,4,0,0,0,0,0,0,14,3,3,3,0,0,3,3,14,0,0,0,0,0,0,3,0,0,0,0,6,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,16,0,0,0,0,16,16,0,0,0,0,0,1],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,15,0,16] >;
C42.49C23 in GAP, Magma, Sage, TeX
C_4^2._{49}C_2^3 % in TeX
G:=Group("C4^2.49C2^3"); // GroupNames label
G:=SmallGroup(128,2046);
// by ID
G=gap.SmallGroup(128,2046);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,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=a^2*c,e*d*e=b^2*d>;
// generators/relations
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