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C₂₉ #88

order	29
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₄×C₇ (order 28) · Inn(G) = C₁ · Out(G) = C₄×C₇
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₂₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

C₂₉

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 29

A minimal faithful permutation representation — 1 generator on 29 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)

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

Relationships

center of	D₃×C₂₉, D₅×C₂₉, A₄×C₂₉, …
derived subgroup of	D₂₉, Dic₂₉, #1181, …
Frattini subgroup of	C₈₄₁, D₈₄₁, C₂×C₈₄₁, …
Fitting subgroup of	D₂₉, #1181, C₂₉⋊C₇, …
socle of	D₂₉, #1181, C₂₉⋊C₇, …
maximal subgroup of	D₂₉, C₂×C₂₉, C₃×C₂₉, …
maximal normal subgroup of	D₂₉, C₂×C₂₉, C₃×C₂₉, …
maximal quotient of	C₂×C₂₉, C₃×C₂₉, C₅×C₂₉, …
outer automorphism group of	D₅₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29, 1);