C₂₉×#3021 #16816735

order	3712
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₂₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂₉×#3021

Subgroups

	subgroup	G/H
center, socle	C₂×C₂₉	C₂×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

736 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 212 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) = 464

A minimal faithful permutation representation — 4 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)(31,36)(33,34)(37,44)(38,41)(39,42)(43,45);(30,31)(32,37)(33,36)(34,41)(35,38)(39,43)(40,45)(42,44);(30,32)(31,37)(33,39)(34,42)(35,40)(36,43)(38,45)(41,44);(30,33,35,34)(31,36,38,41)(32,39,40,42)(37,43,45,44)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#13353, C₂₉×#13354, C₂₉×#13360, …
maximal normal subgroup of	C₂₉×#13353, C₂₉×#13354, C₂₉×#13360, …
maximal quotient of	C₂₉×#18135, C₂₉×#18141, C₂₉×#18145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 1753);