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D805 #13108014

order1610
isoclinismisoclinic class of D805
automorphismAut(G) order 425040
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD805
unambiguous semidirect productC5⋊D161
unambiguous semidirect productC7⋊D115
unambiguous semidirect productC23⋊D35

Splittings

Fitting, socle, derived, HallC5×C7×C23 ⋊ C2
HallC7×C23 ⋊ D5
HallC5×C23 ⋊ D7
HallC5×C7 ⋊ D23
SylowC23 ⋊ D35
SylowC7 ⋊ D115
SylowC5 ⋊ D161

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C23C2
LCSG ▷ C5×C7×C23
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 805

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,3,4,5)(6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35);(2,5)(3,4)(7,12)(8,11)(9,10)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Polycyclic presentation:

2,5,7,23|b^a=b^4,c^a=c^6,d^a=d^22

Conjugacy classes & character table

404 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD23×D35, D7×D115, D5×D161, …
maximal normal subgroup ofD23×D35, D7×D115, D5×D161, …
maximal quotient ofDic805, C2×D805, C3×D805, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1610, 7);