p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊7SD16, C42.460C23, C4.402+ 1+4, D42.4C2, (C8×D4)⋊31C2, C8⋊8D4⋊34C2, C4⋊C8⋊78C22, C4⋊C4.259D4, (C4×C8)⋊45C22, D4⋊3Q8⋊2C2, C4⋊Q8⋊21C22, D4○3(D4⋊C4), C4⋊SD16⋊42C2, D4⋊2Q8⋊39C2, (C4×SD16)⋊37C2, (C2×D4).351D4, C2.44(D4○D8), C4.4D8⋊29C2, C22⋊C4.99D4, (C4×Q8)⋊24C22, C4.44(C2×SD16), C4.Q8⋊36C22, D4.16(C4○D4), C22⋊SD16⋊34C2, C4⋊C4.399C23, C22⋊C8⋊70C22, (C2×C8).346C23, (C2×C4).487C24, (C22×C8)⋊39C22, C23.470(C2×D4), C22⋊Q8⋊15C22, D4⋊C4⋊48C22, C22.5(C2×SD16), Q8⋊C4⋊62C22, (C2×SD16)⋊50C22, (C4×D4).329C22, (C2×D4).220C23, C4⋊D4.70C22, C4⋊1D4.83C22, (C2×Q8).204C23, C2.123(D4⋊5D4), C2.28(C22×SD16), C23.46D4⋊31C2, C22.747(C22×D4), (C22×C4).1131C23, (C22×D4).406C22, (C2×C4⋊C4)⋊56C22, C4.212(C2×C4○D4), (C2×C4).164(C2×D4), (C2×D4⋊C4)⋊40C2, SmallGroup(128,2027)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊7SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd=c3 >
Subgroups: 552 in 231 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊1D4, C4⋊Q8, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×D4, C2×D4⋊C4, C8×D4, C4×SD16, C22⋊SD16, C4⋊SD16, C8⋊8D4, D4⋊2Q8, C23.46D4, C4.4D8, D42, D4⋊3Q8, D4⋊7SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C24, C2×SD16, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C22×SD16, D4○D8, D4⋊7SD16
►On 32 points
(1 9 30 19)(2 10 31 20)(3 11 32 21)(4 12 25 22)(5 13 26 23)(6 14 27 24)(7 15 28 17)(8 16 29 18)
(1 23)(2 14)(3 17)(4 16)(5 19)(6 10)(7 21)(8 12)(9 26)(11 28)(13 30)(15 32)(18 25)(20 27)(22 29)(24 31)
(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 26)(2 29)(3 32)(4 27)(5 30)(6 25)(7 28)(8 31)(9 23)(10 18)(11 21)(12 24)(13 19)(14 22)(15 17)(16 20)
G:=sub<Sym(32)| (1,9,30,19)(2,10,31,20)(3,11,32,21)(4,12,25,22)(5,13,26,23)(6,14,27,24)(7,15,28,17)(8,16,29,18), (1,23)(2,14)(3,17)(4,16)(5,19)(6,10)(7,21)(8,12)(9,26)(11,28)(13,30)(15,32)(18,25)(20,27)(22,29)(24,31), (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), (1,26)(2,29)(3,32)(4,27)(5,30)(6,25)(7,28)(8,31)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20)>;
G:=Group( (1,9,30,19)(2,10,31,20)(3,11,32,21)(4,12,25,22)(5,13,26,23)(6,14,27,24)(7,15,28,17)(8,16,29,18), (1,23)(2,14)(3,17)(4,16)(5,19)(6,10)(7,21)(8,12)(9,26)(11,28)(13,30)(15,32)(18,25)(20,27)(22,29)(24,31), (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), (1,26)(2,29)(3,32)(4,27)(5,30)(6,25)(7,28)(8,31)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20) );
G=PermutationGroup([[(1,9,30,19),(2,10,31,20),(3,11,32,21),(4,12,25,22),(5,13,26,23),(6,14,27,24),(7,15,28,17),(8,16,29,18)], [(1,23),(2,14),(3,17),(4,16),(5,19),(6,10),(7,21),(8,12),(9,26),(11,28),(13,30),(15,32),(18,25),(20,27),(22,29),(24,31)], [(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)], [(1,26),(2,29),(3,32),(4,27),(5,30),(6,25),(7,28),(8,31),(9,23),(10,18),(11,21),(12,24),(13,19),(14,22),(15,17),(16,20)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 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 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | SD16 | C4○D4 | 2+ 1+4 | D4○D8 |
| kernel | D4⋊7SD16 | C2×D4⋊C4 | C8×D4 | C4×SD16 | C22⋊SD16 | C4⋊SD16 | C8⋊8D4 | D4⋊2Q8 | C23.46D4 | C4.4D8 | D42 | D4⋊3Q8 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | D4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 4 | 1 | 2 |
Matrix representation of D4⋊7SD16 ►in GL4(𝔽17) generated by
| 1 | 2 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 2 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 9 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 5 | 5 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,16,0,0,2,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,2,16,0,0,0,0,1,0,0,0,0,1],[13,4,0,0,9,4,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;
D4⋊7SD16 in GAP, Magma, Sage, TeX
D_4\rtimes_7{\rm SD}_{16} % in TeX
G:=Group("D4:7SD16"); // GroupNames label
G:=SmallGroup(128,2027);
// by ID
G=gap.SmallGroup(128,2027);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
// generators/relations