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C₃×C₂₉ #760

order	87
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄×C₇ (order 56) · Inn(G) = C₁ · Out(G) = C₂×C₄×C₇
holomorph	G ⋊ Aut(G) = D₃×AGL(1,𝔽₂₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₈₇
abelian	C₃×C₂₉

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) = 32
transitive degree	μ_t(G) = 87

A minimal faithful permutation representation — 1 generator on 32 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)

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

center of	C₃×D₃×C₂₉, H(3,𝔽₃)×C₂₉, 3₋¹⁺²×C₂₉, …
derived subgroup of	D₈₇, Dic₈₇, C₃⋊#1181, …
Frattini subgroup of	C₂×C₉×C₈₄₁, D₉×C₈₄₁, C₉×D₈₄₁, …
Fitting subgroup of	D₃×C₂₉, C₃×D₂₉, D₈₇, …
socle of	D₃×C₂₉, C₃×D₂₉, D₈₇, …
maximal subgroup of	D₃×C₂₉, C₃×D₂₉, D₈₇, …
maximal normal subgroup of	D₃×C₂₉, C₃×D₂₉, D₈₇, …
maximal quotient of	C₂×C₃×C₂₉, C₉×C₂₉, C₃²×C₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(87, 1);