Finite Group Databasebeta

← Prev (#66305) Next (#66307) →

Cayley diagram

D161 #66306

order322
isoclinismisoclinic class of D161
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD161
unambiguous semidirect productC7⋊D23
unambiguous semidirect productC23⋊D7

Splittings

Fitting, socle, derived, HallC7×C23 ⋊ C2
SylowC23 ⋊ D7
SylowC7 ⋊ D23

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C23C2
LCSG ▷ C7×C23
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 161

A minimal faithful permutation representation — 2 generators on 30 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);(2,7)(3,6)(4,5)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)

Polycyclic presentation:

2,7,23|b^a=b^6,c^a=c^22

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×D23, C2×D161, ΓL(1,𝔽8)⋊D23, …
maximal normal subgroup ofD7×D23, C2×D161, ΓL(1,𝔽8)⋊D23, …
maximal quotient ofDic161, C2×D161, C3×D161, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(322, 3);