p-group, metabelian, nilpotent (class 3), monomial
Aliases: SD16:10D4, C42.449C23, C4.1362+ 1+4, C2.66D42, (C8xD4):11C2, C8:8D4:7C2, C8.84(C2xD4), C22:D8:8C2, D4:D4:6C2, D4:5D4:7C2, C4:C8:69C22, C4:C4.255D4, (C4xC8):19C22, Q8:5D4:6C2, (C4xSD16):9C2, D4.28(C2xD4), C22:C4o2SD16, Q8.26(C2xD4), D4.7D4:5C2, D4.2D4:6C2, (C2xD4).228D4, C8.12D4:8C2, C22:3(C4oD8), C22:Q16:7C2, Q8.D4:5C2, (C4xQ8):23C22, C4.96(C22xD4), C4.Q8:70C22, C4:C4.221C23, C22:C8:62C22, (C2xC4).480C24, (C2xC8).179C23, (C22xC8):19C22, C22:C4.193D4, (C2xQ16):48C22, (C2xD8).34C22, C23.466(C2xD4), C22:Q8:14C22, D4:C4:47C22, C2.64(D4oSD16), Q8:C4:72C22, (C22xSD16):29C2, (C2xSD16):81C22, (C4xD4).324C22, (C2xD4).215C23, C4:D4.66C22, C4.4D4:16C22, (C2xQ8).201C23, C22.740(C22xD4), (C22xC4).1124C23, (C22xD4).404C22, (C22xQ8).334C22, (C2xC4oD8):10C2, C2.53(C2xC4oD8), C22:C4o(C2xSD16), (C2xC4).918(C2xD4), (C2xC4oD4):18C22, SmallGroup(128,2014)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for SD16:10D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, ad=da, cbc-1=dbd=a4b, dcd=c-1 >
Subgroups: 544 in 252 conjugacy classes, 96 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, D8, SD16, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C24, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C4.Q8, C2xC22:C4, C4xD4, C4xD4, C4xQ8, C22wrC2, C4:D4, C4:D4, C22:Q8, C22:Q8, C22.D4, C4.4D4, C4.4D4, C22xC8, C2xD8, C2xSD16, C2xSD16, C2xQ16, C4oD8, C22xD4, C22xQ8, C2xC4oD4, C8xD4, C4xSD16, C22:D8, D4:D4, C22:Q16, D4.7D4, D4.2D4, Q8.D4, C8:8D4, C8.12D4, D4:5D4, Q8:5D4, C22xSD16, C2xC4oD8, SD16:10D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4oD8, C22xD4, 2+ 1+4, D42, C2xC4oD8, D4oSD16, SD16:10D4
►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 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 16 20 26)(2 9 21 27)(3 10 22 28)(4 11 23 29)(5 12 24 30)(6 13 17 31)(7 14 18 32)(8 15 19 25)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
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), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,16,20,26)(2,9,21,27)(3,10,22,28)(4,11,23,29)(5,12,24,30)(6,13,17,31)(7,14,18,32)(8,15,19,25), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)>;
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), (1,20)(2,23)(3,18)(4,21)(5,24)(6,19)(7,22)(8,17)(9,25)(10,28)(11,31)(12,26)(13,29)(14,32)(15,27)(16,30), (1,16,20,26)(2,9,21,27)(3,10,22,28)(4,11,23,29)(5,12,24,30)(6,13,17,31)(7,14,18,32)(8,15,19,25), (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20) );
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)], [(1,20),(2,23),(3,18),(4,21),(5,24),(6,19),(7,22),(8,17),(9,25),(10,28),(11,31),(12,26),(13,29),(14,32),(15,27),(16,30)], [(1,16,20,26),(2,9,21,27),(3,10,22,28),(4,11,23,29),(5,12,24,30),(6,13,17,31),(7,14,18,32),(8,15,19,25)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | C4oD8 | 2+ 1+4 | D4oSD16 |
| kernel | SD16:10D4 | C8xD4 | C4xSD16 | C22:D8 | D4:D4 | C22:Q16 | D4.7D4 | D4.2D4 | Q8.D4 | C8:8D4 | C8.12D4 | D4:5D4 | Q8:5D4 | C22xSD16 | C2xC4oD8 | C22:C4 | C4:C4 | SD16 | C2xD4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 1 | 8 | 1 | 2 |
Matrix representation of SD16:10D4 ►in GL4(F17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 5 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
| 16 | 1 | 0 | 0 |
| 15 | 1 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 13 | 13 |
| 16 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 13 | 13 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,7,5,0,0,7,0],[16,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[16,15,0,0,1,1,0,0,0,0,4,13,0,0,8,13],[16,0,0,0,1,1,0,0,0,0,4,13,0,0,8,13] >;
SD16:10D4 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes_{10}D_4 % in TeX
G:=Group("SD16:10D4"); // GroupNames label
G:=SmallGroup(128,2014);
// by ID
G=gap.SmallGroup(128,2014);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2019,346,2804,1411,375,172]);
// 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,a*d=d*a,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations