p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).38D4, M4(2).12C23, C8.26(C2×D4), C4○D4.21D4, D4.16(C2×D4), Q8.16(C2×D4), D4.3D4⋊2C2, (C2×D4).152D4, D4.5D4⋊4C2, Q8○M4(2)⋊2C2, (C2×C4).20C24, (C2×C8).23C23, (C2×Q8).128D4, C8○D4.5C22, C8.C4⋊4C22, (C2×Q16)⋊19C22, C4○D4.32C23, (C2×D4).74C23, C4.167(C22×D4), C8⋊C22.5C22, (C2×Q8).62C23, C4.178(C4⋊D4), (C2×SD16)⋊14C22, D8⋊C22.8C2, C4.D4⋊16C22, M4(2).C4⋊14C2, C8.C22.4C22, C23.197(C4○D4), C4.10D4⋊16C22, C22.11(C4⋊D4), (C22×C4).288C23, (C2×M4(2)).63C22, (C22×Q8).287C22, M4(2).8C22⋊6C2, (C2×C4).478(C2×D4), C2.91(C2×C4⋊D4), (C2×C8.C22)⋊21C2, C22.23(C2×C4○D4), (C2×C4).480(C4○D4), (C2×C4.10D4)⋊12C2, (C2×C4○D4).133C22, SmallGroup(128,1801)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).38D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 396 in 223 conjugacy classes, 98 normal (36 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.D4, C4.10D4, C8.C4, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×SD16, C2×SD16, C2×Q16, C2×Q16, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C2×C4.10D4, M4(2).8C22, M4(2).C4, D4.3D4, D4.5D4, Q8○M4(2), C2×C8.C22, D8⋊C22, M4(2).38D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4, M4(2).38D4
Character table of M4(2).38D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | -2 | 0 | -2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○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 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)
(1 26 3 32 5 30 7 28)(2 25 4 31 6 29 8 27)(9 22 11 20 13 18 15 24)(10 21 12 19 14 17 16 23)
(1 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 30 21 26)(18 29 22 25)(19 28 23 32)(20 27 24 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), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32), (1,26,3,32,5,30,7,28)(2,25,4,31,6,29,8,27)(9,22,11,20,13,18,15,24)(10,21,12,19,14,17,16,23), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,30,21,26)(18,29,22,25)(19,28,23,32)(20,27,24,31)>;
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), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,29)(18,26)(19,31)(20,28)(21,25)(22,30)(23,27)(24,32), (1,26,3,32,5,30,7,28)(2,25,4,31,6,29,8,27)(9,22,11,20,13,18,15,24)(10,21,12,19,14,17,16,23), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,30,21,26)(18,29,22,25)(19,28,23,32)(20,27,24,31) );
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)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16),(17,29),(18,26),(19,31),(20,28),(21,25),(22,30),(23,27),(24,32)], [(1,26,3,32,5,30,7,28),(2,25,4,31,6,29,8,27),(9,22,11,20,13,18,15,24),(10,21,12,19,14,17,16,23)], [(1,12,5,16),(2,11,6,15),(3,10,7,14),(4,9,8,13),(17,30,21,26),(18,29,22,25),(19,28,23,32),(20,27,24,31)]])
Matrix representation of M4(2).38D4 ►in GL8(𝔽17)
| 2 | 15 | 2 | 2 | 0 | 0 | 0 | 0 |
| 2 | 2 | 15 | 2 | 0 | 0 | 0 | 0 |
| 15 | 15 | 15 | 2 | 0 | 0 | 0 | 0 |
| 2 | 15 | 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 15 | 2 | 2 |
| 0 | 0 | 0 | 0 | 2 | 2 | 15 | 2 |
| 0 | 0 | 0 | 0 | 15 | 15 | 15 | 2 |
| 0 | 0 | 0 | 0 | 2 | 15 | 15 | 15 |
| 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 | 2 | 2 |
| 0 | 0 | 0 | 0 | 2 | 2 | 2 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 15 | 2 |
| 0 | 0 | 0 | 0 | 2 | 15 | 2 | 2 |
| 2 | 2 | 2 | 15 | 0 | 0 | 0 | 0 |
| 2 | 15 | 15 | 15 | 0 | 0 | 0 | 0 |
| 2 | 15 | 2 | 2 | 0 | 0 | 0 | 0 |
| 15 | 15 | 2 | 15 | 0 | 0 | 0 | 0 |
| 2 | 2 | 2 | 15 | 0 | 0 | 0 | 0 |
| 2 | 15 | 15 | 15 | 0 | 0 | 0 | 0 |
| 2 | 15 | 2 | 2 | 0 | 0 | 0 | 0 |
| 15 | 15 | 2 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 | 2 | 2 |
| 0 | 0 | 0 | 0 | 2 | 2 | 2 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 15 | 2 |
| 0 | 0 | 0 | 0 | 2 | 15 | 2 | 2 |
G:=sub<GL(8,GF(17))| [2,2,15,2,0,0,0,0,15,2,15,15,0,0,0,0,2,15,15,15,0,0,0,0,2,2,2,15,0,0,0,0,0,0,0,0,2,2,15,2,0,0,0,0,15,2,15,15,0,0,0,0,2,15,15,15,0,0,0,0,2,2,2,15],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,2,2,2,15,0,0,0,0,2,15,15,15,0,0,0,0,2,15,2,2,0,0,0,0,15,15,2,15,15,2,2,2,0,0,0,0,2,2,2,15,0,0,0,0,2,2,15,2,0,0,0,0,2,15,2,2,0,0,0,0],[2,2,2,15,0,0,0,0,2,15,15,15,0,0,0,0,2,15,2,2,0,0,0,0,15,15,2,15,0,0,0,0,0,0,0,0,15,2,2,2,0,0,0,0,2,2,2,15,0,0,0,0,2,2,15,2,0,0,0,0,2,15,2,2] >;
M4(2).38D4 in GAP, Magma, Sage, TeX
M_4(2)._{38}D_4 % in TeX
G:=Group("M4(2).38D4"); // GroupNames label
G:=SmallGroup(128,1801);
// by ID
G=gap.SmallGroup(128,1801);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,2804,172,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
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