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G = D4○C16order 64 = 26

Central product of D4 and C16

p-group, metabelian, nilpotent (class 2), monomial

Aliases: D4C16, Q8C16, D4.2C8, Q8.2C8, C16M5(2), C16M4(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

C1C2 — D4○C16
C1C2C4C8C2×C8C8○D4 — D4○C16
C1C2 — D4○C16
C1C16 — D4○C16
C1C2C2C2C2C4C4C8 — D4○C16

Generators and relations for D4○C16
 G = < a,b,c | a4=b2=1, c8=a2, bab=a-1, ac=ca, bc=cb >

2C2
2C2
2C2

Smallest permutation representation of D4○C16
On 32 points
Generators in S32
(1 20 9 28)(2 21 10 29)(3 22 11 30)(4 23 12 31)(5 24 13 32)(6 25 14 17)(7 26 15 18)(8 27 16 19)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)
(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)| (1,20,9,28)(2,21,10,29)(3,22,11,30)(4,23,12,31)(5,24,13,32)(6,25,14,17)(7,26,15,18)(8,27,16,19), (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27), (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( (1,20,9,28)(2,21,10,29)(3,22,11,30)(4,23,12,31)(5,24,13,32)(6,25,14,17)(7,26,15,18)(8,27,16,19), (1,28)(2,29)(3,30)(4,31)(5,32)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27), (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([(1,20,9,28),(2,21,10,29),(3,22,11,30),(4,23,12,31),(5,24,13,32),(6,25,14,17),(7,26,15,18),(8,27,16,19)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27)], [(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  C164Q8
 (Cp×D4).C8: (C2×D4).5C8  D4×C16  C169D4  C166D4  C24.78C23  C40.70C23  C5⋊C16.C22  C56.70C23 ...
 C4p.(C2×C8): C162M5(2)  D12.4C8  C16.12D6  D20.6C8  D20.5C8  Dic10.C8  D28.4C8  C16.12D14 ...

40 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D8E···8J16A···16H16I···16T
order122224444488888···816···1616···16
size112221122211112···21···12···2

40 irreducible representations

dim111111112
type++++
imageC1C2C2C2C4C4C8C8D4○C16
kernelD4○C16C2×C16M5(2)C8○D4M4(2)C4○D4D4Q8C1
# reps1331621248

Matrix representation of D4○C16 in GL2(𝔽17) generated by

115
116
,
115
016
,
70
07
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

Export

Subgroup lattice of D4○C16 in TeX

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