p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.11C23, M4(2).4C23, 2- 1+4.8C22, C4⋊Q8⋊4C22, C4≀C2⋊2C22, C4○D4.18D4, D4.54(C2×D4), Q8.54(C2×D4), C4.87C22≀C2, (C2×D4).300D4, D4.8D4⋊3C2, C8⋊C22⋊8C22, (C2×C4).12C24, (C2×Q8).235D4, C4○D4.7C23, C4.57(C22×D4), D4.10D4⋊3C2, D8⋊C22⋊7C2, (C2×D4).36C23, C4.4D4⋊2C22, C23.237(C2×D4), (C22×C4).111D4, C8.C22⋊9C22, (C2×Q8).28C23, C42⋊C22⋊5C2, C22.61C22≀C2, C4.10D4⋊9C22, (C2×2- 1+4)⋊4C2, C22.36(C22×D4), (C22×C4).282C23, C23.38C23⋊5C2, C42⋊C2.99C22, (C2×M4(2)).46C22, (C22×Q8).266C22, (C2×C4).25(C2×D4), C2.57(C2×C22≀C2), (C2×C4.10D4)⋊9C2, (C2×C4○D4).108C22, SmallGroup(128,1752)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).C23
G = < a,b,c,d,e | a8=b2=c2=1, d2=a4, e2=a6, bab=a5, cac=a3, dad-1=a5b, ae=ea, cbc=ebe-1=a4b, bd=db, dcd-1=a4c, ece-1=a6c, ede-1=a4bd >
Subgroups: 636 in 349 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.10D4, C4≀C2, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2- 1+4, 2- 1+4, C2×C4.10D4, C42⋊C22, D4.8D4, D4.10D4, C23.38C23, D8⋊C22, C2×2- 1+4, M4(2).C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2, M4(2).C23
Character table of M4(2).C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 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 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | -2 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ26 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ27 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ28 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, 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 30)(2 27)(3 32)(4 29)(5 26)(6 31)(7 28)(8 25)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)
(1 17)(2 20)(3 23)(4 18)(5 21)(6 24)(7 19)(8 22)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 17 5 21)(2 13 6 9)(3 23 7 19)(4 11 8 15)(10 28 14 32)(12 26 16 30)(18 27 22 31)(20 25 24 29)
(1 29 7 27 5 25 3 31)(2 30 8 28 6 26 4 32)(9 21 15 19 13 17 11 23)(10 22 16 20 14 18 12 24)
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), (1,30)(2,27)(3,32)(4,29)(5,26)(6,31)(7,28)(8,25)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,17)(2,20)(3,23)(4,18)(5,21)(6,24)(7,19)(8,22)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,17,5,21)(2,13,6,9)(3,23,7,19)(4,11,8,15)(10,28,14,32)(12,26,16,30)(18,27,22,31)(20,25,24,29), (1,29,7,27,5,25,3,31)(2,30,8,28,6,26,4,32)(9,21,15,19,13,17,11,23)(10,22,16,20,14,18,12,24)>;
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), (1,30)(2,27)(3,32)(4,29)(5,26)(6,31)(7,28)(8,25)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,17)(2,20)(3,23)(4,18)(5,21)(6,24)(7,19)(8,22)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,17,5,21)(2,13,6,9)(3,23,7,19)(4,11,8,15)(10,28,14,32)(12,26,16,30)(18,27,22,31)(20,25,24,29), (1,29,7,27,5,25,3,31)(2,30,8,28,6,26,4,32)(9,21,15,19,13,17,11,23)(10,22,16,20,14,18,12,24) );
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)], [(1,30),(2,27),(3,32),(4,29),(5,26),(6,31),(7,28),(8,25),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)], [(1,17),(2,20),(3,23),(4,18),(5,21),(6,24),(7,19),(8,22),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30)], [(1,17,5,21),(2,13,6,9),(3,23,7,19),(4,11,8,15),(10,28,14,32),(12,26,16,30),(18,27,22,31),(20,25,24,29)], [(1,29,7,27,5,25,3,31),(2,30,8,28,6,26,4,32),(9,21,15,19,13,17,11,23),(10,22,16,20,14,18,12,24)]])
Matrix representation of M4(2).C23 ►in GL8(𝔽17)
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 | 0 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 4 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 13 | 0 | 0 |
| 13 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 4 |
| 0 | 0 | 4 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 4 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,0,0,0,4,4,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,13,0,0,0,0,0,0,0,4,0,0,13,13,0,0,0,0,0,0,0,4,0,0,0,0,13,13,0,0,0,0,0,0,0,4,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,16],[4,4,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,13,13,0,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,9,13,0,0,0,0,4,4,0,0,0,0,0,0,9,13,0,0],[13,0,0,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,0,0,0,0,4,4,0,0,0,0,0,0,9,13,0,0,0,0,4,4,0,0,0,0,0,0,9,13,0,0,4,4,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,0,13,13,0,0,0,0,0,0,8,4,0,0,0,0] >;
M4(2).C23 in GAP, Magma, Sage, TeX
M_4(2).C_2^3
% in TeX
G:=Group("M4(2).C2^3"); // GroupNames label
G:=SmallGroup(128,1752);
// by ID
G=gap.SmallGroup(128,1752);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2019,2804,1411,718,172,2028]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=c^2=1,d^2=a^4,e^2=a^6,b*a*b=a^5,c*a*c=a^3,d*a*d^-1=a^5*b,a*e=e*a,c*b*c=e*b*e^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c,e*c*e^-1=a^6*c,e*d*e^-1=a^4*b*d>;
// generators/relations
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