p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4○C16, Q8○C16, D4.2C8, Q8.2C8, C16○M5(2), C16○M4(2), M5(2)⋊7C2, C16.7C22, C8.17C23, M4(2).4C4, (C2×C16)⋊9C2, C4.5(C2×C8), C16○(C4○D4), C16○(C8○D4), C8.13(C2×C4), C8○D4.3C2, C4○D4.3C4, C22.1(C2×C8), C2.7(C22×C8), C4.36(C22×C4), (C2×C8).105C22, (C2×C4).48(C2×C4), SmallGroup(64,185)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4○C16
G = < a,b,c | a4=b2=1, c8=a2, bab=a-1, ac=ca, bc=cb >
Smallest permutation representation of D4○C16
►On 32 points
(1 32 9 24)(2 17 10 25)(3 18 11 26)(4 19 12 27)(5 20 13 28)(6 21 14 29)(7 22 15 30)(8 23 16 31)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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)
G:=sub<Sym(32)| (1,32,9,24)(2,17,10,25)(3,18,11,26)(4,19,12,27)(5,20,13,28)(6,21,14,29)(7,22,15,30)(8,23,16,31), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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)>;
G:=Group( (1,32,9,24)(2,17,10,25)(3,18,11,26)(4,19,12,27)(5,20,13,28)(6,21,14,29)(7,22,15,30)(8,23,16,31), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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) );
G=PermutationGroup([[(1,32,9,24),(2,17,10,25),(3,18,11,26),(4,19,12,27),(5,20,13,28),(6,21,14,29),(7,22,15,30),(8,23,16,31)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(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)]])
D4○C16 is a maximal subgroup of
D4.C16 D4.3D8 D4.4D8 D4.5D8 C16.A4 C5⋊C16.C22
D4p.C8: C16○D8 D8.C8 D12.4C8 C16.12D6 D20.6C8 D20.5C8 Dic10.C8 D28.4C8 ...
C8p.C23: D4○C32 Q8○M5(2) D4○D16 D4○SD32 Q8○D16 C24.78C23 C40.70C23 C56.70C23 ...
D4○C16 is a maximal quotient of
C4⋊C4.7C8 C8.12M4(2) Q8×C16 C16⋊4Q8
(Cp×D4).C8: (C2×D4).5C8 D4×C16 C16⋊9D4 C16⋊6D4 C24.78C23 C40.70C23 C5⋊C16.C22 C56.70C23 ...
C4p.(C2×C8): C16○2M5(2) D12.4C8 C16.12D6 D20.6C8 D20.5C8 Dic10.C8 D28.4C8 C16.12D14 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 16A | ··· | 16H | 16I | ··· | 16T |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4○C16 |
| kernel | D4○C16 | C2×C16 | M5(2) | C8○D4 | M4(2) | C4○D4 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 12 | 4 | 8 |
Matrix representation of D4○C16 ►in GL2(𝔽17) generated by
| 1 | 15 |
| 1 | 16 |
| 1 | 15 |
| 0 | 16 |
| 7 | 0 |
| 0 | 7 |
G:=sub<GL(2,GF(17))| [1,1,15,16],[1,0,15,16],[7,0,0,7] >;
D4○C16 in GAP, Magma, Sage, TeX
D_4\circ C_{16} % in TeX
G:=Group("D4oC16"); // GroupNames label
G:=SmallGroup(64,185);
// by ID
G=gap.SmallGroup(64,185);
# by ID
G:=PCGroup([6,-2,2,2,-2,-2,-2,48,332,69,88]);
// Polycyclic
G:=Group<a,b,c|a^4=b^2=1,c^8=a^2,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations
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