D₈₆₆₅ #31773168

order	17330
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₆₆₅
unambiguous semidirect product	C₅⋊D₁₇₃₃
unambiguous semidirect product	C₁₇₃₃⋊D₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₇₃₃ ⋊ C₂
Sylow	C₁₇₃₃ ⋊ D₅
Sylow	C₅ ⋊ D₁₇₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₇₃₃	C₂
LCS	G ▷ C₅×C₁₇₃₃
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1738
transitive degree	μ_t(G) = 8665

A minimal faithful permutation representation — 2 generators on 1738 points:

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Polycyclic presentation:

2,5,1733|b^a=b^4,c^a=c^1732

Conjugacy classes & character table

4334 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#46831714, #46831715, D₅×D₁₇₃₃, …
maximal normal subgroup of	#46831714, #46831715, D₅×D₁₇₃₃, …
maximal quotient of	Dic₈₆₆₅, C₂×D₈₆₆₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17330, 3);

D8665 #31773168