C₂₉×#1679 #16815393

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂₉×#1679

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂×D₄×C₂₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

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

normal subgroups only

This group has 166 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) = 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)(30,31,34,36)(32,37,35,33)(38,39)(40,42)(41,43)(44,45);(30,32,33,31,34,35,37,36)(39,42)(43,44);(39,43)(41,45)(42,44)

Conjugacy classes & character table

841 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1679, C₂₉×#19110, C₂₉×#19124, …
maximal normal subgroup of	C₂×C₂₉×#1679, C₂₉×#19110, C₂₉×#19124, …
maximal quotient of	C₂₉×#8646, C₂₉×#9217, C₂₉×#9219, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 411);