p-group, metabelian, nilpotent (class 4), monomial
Aliases: D4○D16, Q8○Q32, D4.12D8, Q8.12D8, D16⋊5C22, C8.16C24, C16.3C23, Q32⋊7C22, D8.5C23, SD32⋊4C22, Q16.5C23, M4(2).21D4, M5(2)⋊8C22, D4○D8⋊5C2, C4○D16⋊4C2, D4○C16⋊3C2, C8.15(C2×D4), C4.49(C2×D8), (C2×D16)⋊13C2, C16⋊C22⋊5C2, (C2×C16)⋊5C22, C4○D4.35D4, C4○D8⋊1C22, C22.6(C2×D8), (C2×D8)⋊33C22, C2.31(C22×D8), C4.22(C22×D4), (C2×C8).294C23, C8○D4.12C22, (C2×C4).184(C2×D4), SmallGroup(128,2147)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 480 in 185 conjugacy classes, 90 normal (11 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], D8 [×6], D8 [×6], SD16 [×6], Q16 [×2], C2×D4 [×12], C4○D4, C4○D4 [×8], C2×C16 [×3], M5(2) [×3], D16 [×9], SD32 [×6], Q32, C8○D4, C2×D8 [×6], C4○D8 [×6], C8⋊C22 [×6], 2+ (1+4) [×2], D4○C16, C2×D16 [×3], C4○D16 [×3], C16⋊C22 [×6], D4○D8 [×2], D4○D16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C22×D8, D4○D16
Generators and relations
G = < a,b,c,d | a4=b2=d2=1, c8=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c7 >
►On 32 points
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 32)(15 31)(16 30)
G:=sub<Sym(32)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,32)(15,31)(16,30)>;
G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,32)(15,31)(16,30) );
G=PermutationGroup([(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,32),(15,31),(16,30)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 16 | 0 | 16 | 0 |
| 0 | 16 | 0 | 15 |
| 1 | 0 | 2 | 0 |
| 0 | 1 | 0 | 1 |
| 16 | 0 | 16 | 0 |
| 4 | 11 | 0 | 0 |
| 6 | 4 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 6 | 4 |
| 3 | 14 | 6 | 11 |
| 14 | 14 | 11 | 11 |
| 0 | 0 | 14 | 3 |
| 0 | 0 | 3 | 3 |
G:=sub<GL(4,GF(17))| [0,1,0,16,16,0,1,0,0,0,0,16,0,0,1,0],[0,1,0,16,16,0,1,0,0,2,0,16,15,0,1,0],[4,6,0,0,11,4,0,0,0,0,4,6,0,0,11,4],[3,14,0,0,14,14,0,0,6,11,14,3,11,11,3,3] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 16A | 16B | 16C | 16D | 16E | ··· | 16J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 | 16 | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | D4○D16 |
| kernel | D4○D16 | D4○C16 | C2×D16 | C4○D16 | C16⋊C22 | D4○D8 | M4(2) | C4○D4 | D4 | Q8 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 3 | 1 | 6 | 2 | 4 |
In GAP, Magma, Sage, TeX
D_4\circ D_{16} % in TeX
G:=Group("D4oD16"); // GroupNames label
G:=SmallGroup(128,2147);
// by ID
G=gap.SmallGroup(128,2147);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,521,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^8=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^7>;
// generators/relations