C₂₉×#1621 #16815335

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1621

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 1856

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)(32,34)(33,36)(35,37)(39,42)(40,44)(43,45);(31,34)(32,37)(35,36)(40,44)(42,45);(38,39,41,43)(40,45,44,42)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1621, C₂₉×#18510, C₂₉×#18516, …
maximal normal subgroup of	C₂×C₂₉×#1621, C₂₉×#18510, C₂₉×#18516, …
maximal quotient of	C₂₉×#7735, C₂₉×#7737, C₂₉×#7740, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 353);