p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16⋊2D4, C42.442C23, C4.1292+ (1+4), C8.7(C2×D4), C8⋊6D4⋊6C2, C2.59(D42), D4⋊5D4⋊5C2, C8⋊2D4⋊22C2, C4⋊C8⋊31C22, C4⋊C4.151D4, (C4×C8)⋊32C22, Q8⋊5D4⋊4C2, D4.24(C2×D4), Q8.22(C2×D4), D4⋊D4⋊39C2, Q8⋊D4⋊18C2, (C4×SD16)⋊20C2, (C2×D4).165D4, C8.D4⋊21C2, (C2×C8).90C23, (C4×Q8)⋊21C22, C4.89(C22×D4), C4.Q8⋊53C22, D4.2D4⋊39C2, C22⋊SD16⋊19C2, D4.7D4⋊39C2, C8.12D4⋊22C2, C4⋊C4.214C23, C4⋊D4⋊11C22, C22⋊C8⋊27C22, (C2×C4).473C24, Q8.D4⋊38C2, C22⋊C4.161D4, (C2×Q16)⋊29C22, (C2×D8).81C22, C23.315(C2×D4), D4⋊C4⋊39C22, C2.61(D4○SD16), Q8⋊C4⋊84C22, (C2×SD16)⋊47C22, (C2×D4).212C23, (C4×D4).147C22, C4.4D4⋊14C22, (C2×Q8).196C23, (C22×Q8)⋊24C22, C22⋊Q8.60C22, (C2×M4(2))⋊25C22, (C22×C4).323C23, C22.733(C22×D4), C2.80(D8⋊C22), (C22×D4).401C22, (C2×C8⋊C22)⋊33C2, (C2×C4).157(C2×D4), (C2×C8.C22)⋊30C2, (C2×C4○D4).188C22, SmallGroup(128,2007)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 544 in 249 conjugacy classes, 94 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×20], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×16], D4 [×2], D4 [×15], Q8 [×2], Q8 [×7], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×4], D8 [×4], SD16 [×4], SD16 [×6], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×6], C24, C4×C8, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2×C22⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2, C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4 [×2], C4.4D4, C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×5], C2×Q16 [×2], C8⋊C22 [×4], C8.C22 [×4], C22×D4, C22×Q8, C2×C4○D4 [×2], C8⋊6D4, C4×SD16, Q8⋊D4, D4⋊D4, C22⋊SD16, D4.7D4, D4.2D4, Q8.D4, C8⋊2D4, C8.D4, C8.12D4, D4⋊5D4, Q8⋊5D4, C2×C8⋊C22, C2×C8.C22, SD16⋊2D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ (1+4), D42, D8⋊C22, D4○SD16, SD16⋊2D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, dad=a5, cbc-1=dbd=a4b, dcd=c-1 >
►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)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 10 19 32)(2 11 20 25)(3 12 21 26)(4 13 22 27)(5 14 23 28)(6 15 24 29)(7 16 17 30)(8 9 18 31)
(1 32)(2 29)(3 26)(4 31)(5 28)(6 25)(7 30)(8 27)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,10,19,32)(2,11,20,25)(3,12,21,26)(4,13,22,27)(5,14,23,28)(6,15,24,29)(7,16,17,30)(8,9,18,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)>;
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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,10,19,32)(2,11,20,25)(3,12,21,26)(4,13,22,27)(5,14,23,28)(6,15,24,29)(7,16,17,30)(8,9,18,31), (1,32)(2,29)(3,26)(4,31)(5,28)(6,25)(7,30)(8,27)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17) );
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)], [(1,19),(2,22),(3,17),(4,20),(5,23),(6,18),(7,21),(8,24),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30)], [(1,10,19,32),(2,11,20,25),(3,12,21,26),(4,13,22,27),(5,14,23,28),(6,15,24,29),(7,16,17,30),(8,9,18,31)], [(1,32),(2,29),(3,26),(4,31),(5,28),(6,25),(7,30),(8,27),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 | 1 | 2 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 16 | 1 | 1 |
| 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 | 16 | 16 | 1 | 2 |
| 0 | 0 | 1 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 13 | 9 |
| 0 | 0 | 0 | 13 | 4 | 4 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 4 | 8 |
| 0 | 0 | 4 | 0 | 13 | 13 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,16,0,0,0,16,1,16,0,0,1,1,0,1,0,0,0,2,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,1,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,2,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,4,4,0,0,0,13,0,4,13,0,0,0,0,13,4,0,0,0,0,9,4],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,4,13,4,0,0,13,0,13,0,0,0,0,0,4,13,0,0,0,0,8,13] >;
Character table of SD16⋊2D4
| 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 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ (1+4) |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ28 | 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 | 2√-2 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ29 | 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 | 2√-2 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
In GAP, Magma, Sage, TeX
SD_{16}\rtimes_2D_4 % in TeX
G:=Group("SD16:2D4"); // GroupNames label
G:=SmallGroup(128,2007);
// by ID
G=gap.SmallGroup(128,2007);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,723,346,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^3,a*c=c*a,d*a*d=a^5,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations