C₂₉×#1600 #16815314

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1600

Subgroups

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

320 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 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)(31,36)(32,33)(37,44)(38,42)(39,45)(41,43);(30,31)(32,36)(33,42)(34,38)(35,39)(37,40)(41,45)(43,44);(30,32,40,43)(31,36,37,44)(33,35,41,34)(38,42,39,45)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1600, C₂₉×#18114, C₂₉×#18126, …
maximal normal subgroup of	C₂×C₂₉×#1600, C₂₉×#18114, C₂₉×#18126, …
maximal quotient of	C₂₉×#7737, C₂₉×#7740, C₂₉×#7742, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 332);