C₂₉×#2027 #16815741

order	3712
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#2027

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂²×D₄×C₂₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

282 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 45 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)(30,31,33,35)(32,36,37,34)(38,39)(40,42)(41,44)(43,45);(30,32,33,37)(31,34,35,36)(40,45)(42,43);(39,42)(40,45)(43,44)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#7430, C₂₉×#12208, C₂₉×#12209, …
maximal normal subgroup of	C₂₉×#7430, C₂₉×#12208, C₂₉×#12209, …
maximal quotient of	C₂₉×#10340, C₂₉×#10343, C₂₉×#10344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 759);