p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.13C23, M4(2).6C23, 2- 1+4.9C22, 2+ 1+4.11C22, C4⋊Q8⋊5C22, C4≀C2⋊4C22, C4○D4.53D4, D4.56(C2×D4), Q8.56(C2×D4), (C2×D4).147D4, D4.9D4⋊4C2, (C2×C4).14C24, (C2×Q8).124D4, C4○D4.9C23, C23.24(C2×D4), C4.59(C22×D4), C4.109C22≀C2, D4.10D4⋊4C2, (C2×D4).38C23, (C22×C4).113D4, C4.4D4⋊4C22, (C2×Q8).30C23, C42⋊C22⋊7C2, C22.24C22≀C2, C4.D4⋊11C22, C8.C22⋊10C22, C2.C25.4C2, (C22×Q8)⋊17C22, C22.38(C22×D4), C42⋊C2⋊11C22, C4.10D4⋊11C22, (C2×M4(2))⋊12C22, (C22×C4).284C23, C23.38C23⋊6C2, M4(2).8C22⋊3C2, (C2×C4).462(C2×D4), C2.59(C2×C22≀C2), (C2×C8.C22)⋊13C2, (C2×C4○D4).110C22, SmallGroup(128,1754)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.13C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=b2, cac=ab=ba, dad-1=a-1b2, eae-1=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=b-1c, ce=ec, ede-1=b2d >
Subgroups: 668 in 352 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), 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.D4, C4.10D4, C4≀C2, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, M4(2).8C22, C42⋊C22, D4.9D4, D4.10D4, C23.38C23, C2×C8.C22, C2.C25, C42.13C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2, C42.13C23
Character table of C42.13C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 0 | 0 | -2 | 0 | -2 | 0 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ26 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | -2 | 0 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ27 | 2 | 2 | -2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ28 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 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 6 15 27)(2 7 16 28)(3 8 13 25)(4 5 14 26)(9 29 21 17)(10 30 22 18)(11 31 23 19)(12 32 24 20)
(1 17)(2 10)(3 31)(4 24)(5 32)(6 21)(7 18)(8 11)(9 27)(12 14)(13 19)(15 29)(16 22)(20 26)(23 25)(28 30)
(1 5 15 26)(2 25 16 8)(3 7 13 28)(4 27 14 6)(9 32 21 20)(10 19 22 31)(11 30 23 18)(12 17 24 29)
(1 11 15 23)(2 24 16 12)(3 9 13 21)(4 22 14 10)(5 18 26 30)(6 31 27 19)(7 20 28 32)(8 29 25 17)
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), (1,6,15,27)(2,7,16,28)(3,8,13,25)(4,5,14,26)(9,29,21,17)(10,30,22,18)(11,31,23,19)(12,32,24,20), (1,17)(2,10)(3,31)(4,24)(5,32)(6,21)(7,18)(8,11)(9,27)(12,14)(13,19)(15,29)(16,22)(20,26)(23,25)(28,30), (1,5,15,26)(2,25,16,8)(3,7,13,28)(4,27,14,6)(9,32,21,20)(10,19,22,31)(11,30,23,18)(12,17,24,29), (1,11,15,23)(2,24,16,12)(3,9,13,21)(4,22,14,10)(5,18,26,30)(6,31,27,19)(7,20,28,32)(8,29,25,17)>;
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), (1,6,15,27)(2,7,16,28)(3,8,13,25)(4,5,14,26)(9,29,21,17)(10,30,22,18)(11,31,23,19)(12,32,24,20), (1,17)(2,10)(3,31)(4,24)(5,32)(6,21)(7,18)(8,11)(9,27)(12,14)(13,19)(15,29)(16,22)(20,26)(23,25)(28,30), (1,5,15,26)(2,25,16,8)(3,7,13,28)(4,27,14,6)(9,32,21,20)(10,19,22,31)(11,30,23,18)(12,17,24,29), (1,11,15,23)(2,24,16,12)(3,9,13,21)(4,22,14,10)(5,18,26,30)(6,31,27,19)(7,20,28,32)(8,29,25,17) );
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)], [(1,6,15,27),(2,7,16,28),(3,8,13,25),(4,5,14,26),(9,29,21,17),(10,30,22,18),(11,31,23,19),(12,32,24,20)], [(1,17),(2,10),(3,31),(4,24),(5,32),(6,21),(7,18),(8,11),(9,27),(12,14),(13,19),(15,29),(16,22),(20,26),(23,25),(28,30)], [(1,5,15,26),(2,25,16,8),(3,7,13,28),(4,27,14,6),(9,32,21,20),(10,19,22,31),(11,30,23,18),(12,17,24,29)], [(1,11,15,23),(2,24,16,12),(3,9,13,21),(4,22,14,10),(5,18,26,30),(6,31,27,19),(7,20,28,32),(8,29,25,17)]])
Matrix representation of C42.13C23 ►in GL8(𝔽17)
| 3 | 13 | 13 | 4 | 10 | 7 | 0 | 10 |
| 3 | 13 | 4 | 13 | 10 | 7 | 0 | 7 |
| 4 | 13 | 3 | 13 | 0 | 7 | 10 | 7 |
| 13 | 4 | 3 | 13 | 0 | 10 | 10 | 7 |
| 13 | 14 | 14 | 3 | 1 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 8 |
| 3 | 14 | 13 | 14 | 0 | 16 | 1 | 16 |
| 16 | 1 | 0 | 0 | 0 | 9 | 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 | 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 | 0 | 16 | 1 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 16 | 1 | 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 16 | 1 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 16 | 1 | 0 | 0 |
| 4 | 13 | 3 | 13 | 0 | 7 | 10 | 7 |
| 13 | 4 | 3 | 13 | 0 | 10 | 10 | 7 |
| 3 | 13 | 13 | 4 | 10 | 7 | 0 | 10 |
| 3 | 13 | 4 | 13 | 10 | 7 | 0 | 7 |
| 3 | 14 | 13 | 14 | 0 | 16 | 1 | 16 |
| 16 | 1 | 0 | 0 | 0 | 9 | 0 | 0 |
| 13 | 14 | 14 | 3 | 1 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 1 | 15 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [3,3,4,13,13,0,3,16,13,13,13,4,14,0,14,1,13,4,3,3,14,1,13,0,4,13,13,13,3,16,14,0,10,10,0,0,1,0,0,0,7,7,7,10,16,0,16,9,0,0,10,10,0,0,1,0,10,7,7,7,1,8,16,0],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,16,0,0,0,0,0,0,0,1,0,0,0,16,16,16,0,0,0,0,0,0,0,1,0,0,0,16,16,0,0,0,0,1,16,0,0,0,16,0,0,15,0,0,0,0,1,1,16,0,0,0,16,0,0,15,0,0,0,0,1,0,0],[4,13,3,3,3,16,13,0,13,4,13,13,14,1,14,0,3,3,13,4,13,0,14,1,13,13,4,13,14,0,3,16,0,0,10,10,0,0,1,0,7,10,7,7,16,9,16,0,10,10,0,0,1,0,0,0,7,7,10,7,16,0,1,8],[0,16,0,0,16,16,0,0,0,1,0,0,0,1,0,0,0,0,0,16,0,0,16,16,0,0,0,1,0,0,0,1,1,1,0,0,0,0,0,0,0,15,0,0,0,16,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,0,0,16] >;
C42.13C23 in GAP, Magma, Sage, TeX
C_4^2._{13}C_2^3 % in TeX
G:=Group("C4^2.13C2^3"); // GroupNames label
G:=SmallGroup(128,1754);
// by ID
G=gap.SmallGroup(128,1754);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,248,2804,1411,718,172,2028]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=e^2=b^2,c*a*c=a*b=b*a,d*a*d^-1=a^-1*b^2,e*a*e^-1=a*b^2,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b^-1*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
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