p-group, metabelian, nilpotent (class 4), monomial
Aliases: D16⋊4C22, C8.15C24, C16.2C23, Q32⋊4C22, D8.4C23, C23.21D8, SD32⋊3C22, Q16.4C23, M5(2)⋊7C22, C4○D16⋊3C2, C4.77(C2×D8), (C2×C4).56D8, C8.58(C2×D4), C4○(C16⋊C22), C16⋊C22⋊7C2, (C2×C16)⋊4C22, C4○(Q32⋊C2), Q32⋊C2⋊7C2, (C2×C8).149D4, C4○D8⋊6C22, (C2×D8)⋊53C22, (C2×M5(2))⋊5C2, C22.26(C2×D8), C4.21(C22×D4), C2.30(C22×D8), (C2×C8).293C23, (C2×Q16)⋊57C22, (C22×C4).534D4, (C22×C8).296C22, (C2×C4○D8)⋊28C2, (C2×C4).660(C2×D4), SmallGroup(128,2146)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D16⋊C22
G = < a,b,c,d | a16=b2=c2=d2=1, bab=a-1, cac=a9, ad=da, cbc=dbd=a8b, cd=dc >
Subgroups: 404 in 182 conjugacy classes, 90 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C16, M5(2), D16, SD32, Q32, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C2×M5(2), C4○D16, C16⋊C22, Q32⋊C2, C2×C4○D8, D16⋊C22
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, C22×D8, D16⋊C22
►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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,17),(14,18),(15,19),(16,20)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | D16⋊C22 |
| kernel | D16⋊C22 | C2×M5(2) | C4○D16 | C16⋊C22 | Q32⋊C2 | C2×C4○D8 | C2×C8 | C22×C4 | C2×C4 | C23 | C1 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 3 | 1 | 6 | 2 | 4 |
Matrix representation of D16⋊C22 ►in GL4(𝔽17) generated by
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 12 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 4 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [0,0,12,5,0,0,5,5,0,4,0,0,4,0,0,0],[0,0,13,0,0,0,0,13,4,0,0,0,0,4,0,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,0,4,0,0,13,0] >;
D16⋊C22 in GAP, Magma, Sage, TeX
D_{16}\rtimes C_2^2 % in TeX
G:=Group("D16:C2^2"); // GroupNames label
G:=SmallGroup(128,2146);
// by ID
G=gap.SmallGroup(128,2146);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,1430,248,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^9,a*d=d*a,c*b*c=d*b*d=a^8*b,c*d=d*c>;
// generators/relations