p-group, metabelian, nilpotent (class 4), monomial
Aliases: D16⋊5C4, Q32⋊5C4, C8.33D8, SD32⋊4C4, C42.135D4, C8○D8⋊2C2, C16.9(C2×C4), C16⋊5C4⋊3C2, D8.5(C2×C4), C2.20(C4×D8), C4.32(C4×D4), C4.93(C2×D8), C4○D16.3C2, (C2×C8).213D4, Q16.5(C2×C4), C8.4Q8⋊4C2, D8.C4⋊6C2, C8.59(C4○D4), C8.42(C22×C4), (C4×C8).161C22, (C2×C8).581C23, (C2×C16).22C22, C4○D8.15C22, C22.2(C4○D8), C8.C4.17C22, (C2×C4).775(C2×D4), SmallGroup(128,911)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D16⋊5C4
G = < a,b,c | a16=b2=c4=1, bab=a-1, cac-1=a9, cbc-1=a4b >
Subgroups: 148 in 72 conjugacy classes, 38 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C8, C2×C4, C2×C4, D4, Q8, C16, C16, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, C4×C8, C4≀C2, C8.C4, C2×C16, D16, SD32, Q32, C8○D4, C4○D8, C16⋊5C4, D8.C4, C8.4Q8, C8○D8, C4○D16, D16⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, D16⋊5C4
►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 18)(2 17)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
(2 10)(4 12)(6 14)(8 16)(17 29 25 21)(18 22 26 30)(19 31 27 23)(20 24 28 32)
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), (1,18)(2,17)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19), (2,10)(4,12)(6,14)(8,16)(17,29,25,21)(18,22,26,30)(19,31,27,23)(20,24,28,32)>;
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), (1,18)(2,17)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19), (2,10)(4,12)(6,14)(8,16)(17,29,25,21)(18,22,26,30)(19,31,27,23)(20,24,28,32) );
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)], [(1,18),(2,17),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19)], [(2,10),(4,12),(6,14),(8,16),(17,29,25,21),(18,22,26,30),(19,31,27,23),(20,24,28,32)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | C4○D4 | C4○D8 | D16⋊5C4 |
| kernel | D16⋊5C4 | C16⋊5C4 | D8.C4 | C8.4Q8 | C8○D8 | C4○D16 | D16 | SD32 | Q32 | C42 | C2×C8 | C8 | C8 | C22 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 4 | 2 | 4 | 4 |
Matrix representation of D16⋊5C4 ►in GL4(𝔽17) generated by
| 0 | 0 | 0 | 15 |
| 0 | 0 | 15 | 0 |
| 0 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [0,0,0,1,0,0,13,0,0,15,0,0,15,0,0,0],[0,0,8,0,0,0,0,8,15,0,0,0,0,15,0,0],[1,0,0,0,0,4,0,0,0,0,13,0,0,0,0,16] >;
D16⋊5C4 in GAP, Magma, Sage, TeX
D_{16}\rtimes_5C_4 % in TeX
G:=Group("D16:5C4"); // GroupNames label
G:=SmallGroup(128,911);
// by ID
G=gap.SmallGroup(128,911);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,1123,570,360,172,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^4*b>;
// generators/relations