D₈₇₀₅ #31775141

order	17410
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₇₀₅
unambiguous semidirect product	C₅⋊D₁₇₄₁
unambiguous semidirect product	C₁₇₄₁⋊D₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₇₄₁ ⋊ C₂
Sylow	C₁₇₄₁ ⋊ D₅
Sylow	C₅ ⋊ D₁₇₄₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₇₄₁	C₂
LCS	G ▷ C₅×C₁₇₄₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1746
transitive degree	μ_t(G) = 8705

A minimal faithful permutation representation — 2 generators on 1746 points:

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Polycyclic presentation:

2,5,1741|b^a=b^4,c^a=c^1740

Conjugacy classes & character table

4354 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#46852593, #46852594, D₅×D₁₇₄₁, …
maximal normal subgroup of	#46852593, #46852594, D₅×D₁₇₄₁, …
maximal quotient of	Dic₈₇₀₅, C₂×D₈₇₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17410, 5);

D8705 #31775141