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D665 #13050600

order1330
isoclinismisoclinic class of D665
automorphismAut(G) order 287280
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD665
unambiguous semidirect productC5⋊D133
unambiguous semidirect productC7⋊D95
unambiguous semidirect productC19⋊D35

Splittings

Fitting, socle, derived, HallC5×C7×C19 ⋊ C2
HallC7×C19 ⋊ D5
HallC5×C19 ⋊ D7
HallC5×C7 ⋊ D19
SylowC19 ⋊ D35
SylowC7 ⋊ D95
SylowC5 ⋊ D133

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C19C2
LCSG ▷ C5×C7×C19
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 665

A minimal faithful permutation representation — 2 generators on 31 points:

(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31);(2,5)(3,4)(7,12)(8,11)(9,10)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)

Polycyclic presentation:

2,5,7,19|b^a=b^4,c^a=c^6,d^a=d^18

Conjugacy classes & character table

334 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD19×D35, D7×D95, D5×D133, …
maximal normal subgroup ofD19×D35, D7×D95, D5×D133, …
maximal quotient ofDic665, C2×D665, C3×D665, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1330, 7);