C₂₉×#1601 #16815315

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

Aliases

direct product

C₂₉×#1601

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂₉×#534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

308 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 68 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)(30,31,33,36)(32,34,37,35)(42,45);(31,34)(32,37)(35,36)(38,39)(40,42)(41,43)(44,45);(38,40)(39,42)(41,44)(43,45)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12120, C₂₉×#12121, C₂₉×#12123, …
maximal normal subgroup of	C₂₉×#12120, C₂₉×#12121, C₂₉×#12123, …
maximal quotient of	C₂₉×#7750, C₂₉×#7753, C₂₉×#7762, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 333);