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G = D4oC16order 64 = 26

Central product of D4 and C16

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

Aliases: D4oC16, Q8oC16, D4.2C8, Q8.2C8, C16oM5(2), C16oM4(2), M5(2):7C2, C16.7C22, C8.17C23, M4(2).4C4, (C2xC16):9C2, C4.5(C2xC8), C16o(C4oD4), C16o(C8oD4), C8.13(C2xC4), C8oD4.3C2, C4oD4.3C4, C22.1(C2xC8), C2.7(C22xC8), C4.36(C22xC4), (C2xC8).105C22, (C2xC4).48(C2xC4), SmallGroup(64,185)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — D4oC16
C1C2C4C8C2xC8C8oD4 — D4oC16
C1C2 — D4oC16
C1C16 — D4oC16
C1C2C2C2C2C4C4C8 — D4oC16

Generators and relations for D4oC16
 G = < a,b,c | a4=b2=1, c8=a2, bab=a-1, ac=ca, bc=cb >

Subgroups: 45 in 42 conjugacy classes, 39 normal (9 characteristic)
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C2xC8, C22xC4, C22xC8, D4oC16
2C2
2C2
2C2

Smallest permutation representation of D4oC16
On 32 points
Generators in S32
(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)]])

D4oC16 is a maximal subgroup of
D4.C16  D4.3D8  D4.4D8  D4.5D8  C16.A4  C5:C16.C22
 D4p.C8: C16oD8  D8.C8  D12.4C8  C16.12D6  D20.6C8  D20.5C8  Dic10.C8  D28.4C8 ...
 C8p.C23: D4oC32  Q8oM5(2)  D4oD16  D4oSD32  Q8oD16  C24.78C23  C40.70C23  C56.70C23 ...
D4oC16 is a maximal quotient of
C4:C4.7C8  C8.12M4(2)  Q8xC16  C16:4Q8
 (CpxD4).C8: (C2xD4).5C8  D4xC16  C16:9D4  C16:6D4  C24.78C23  C40.70C23  C5:C16.C22  C56.70C23 ...
 C4p.(C2xC8): C16o2M5(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++++
imageC1C2C2C2C4C4C8C8D4oC16
kernelD4oC16C2xC16M5(2)C8oD4M4(2)C4oD4D4Q8C1
# reps1331621248

Matrix representation of D4oC16 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] >;

D4oC16 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 D4oC16 in TeX

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