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C₅×C₁₇ #757

order	85
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×C₁₆ (order 64) · Inn(G) = C₁ · Out(G) = C₄×C₁₆
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₁₇)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₈₅
abelian	C₅×C₁₇

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 85

A minimal faithful permutation representation — 1 generator on 22 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

center of	C₅×D₃×C₁₇, C₅×D₅×C₁₇, C₅×A₄×C₁₇, …
derived subgroup of	D₈₅, Dic₈₅, #66772, …
Frattini subgroup of	C₂×C₂₅×C₂₈₉, D₂₅×C₂₈₉, C₂₅×D₂₈₉, …
Fitting subgroup of	D₅×C₁₇, C₅×D₁₇, D₈₅, …
socle of	D₅×C₁₇, C₅×D₁₇, D₈₅, …
maximal subgroup of	D₅×C₁₇, C₅×D₁₇, D₈₅, …
maximal normal subgroup of	D₅×C₁₇, C₅×D₁₇, D₈₅, …
maximal quotient of	C₂×C₅×C₁₇, C₃×C₅×C₁₇, C₁₇×C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(85, 1);