C₂₉×#1827 #16815541

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₂₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂₉×#1827

Subgroups

	subgroup	G/H
center	C₄×C₂₉	#117
commutator	C₂×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
socle	C₂×C₂₉	#381
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₉ ◁ C₂₉×#117 ◁ G

262 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 100 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)(30,31)(32,39,41,35)(33,38)(34,37)(36,40,43,45)(42,44);(30,32)(31,35)(33,40)(34,41)(36,38)(37,39)(42,45)(43,44);(30,32)(31,36,37,43)(33,40)(34,41)(35,38,39,44)(42,45)

Conjugacy classes & character table

928 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12719, C₂₉×#12720, C₂₉×#12721, …
maximal normal subgroup of	C₂₉×#12719, C₂₉×#12720, C₂₉×#12721, …
maximal quotient of	C₂₉×#8730, C₂₉×#8731, C₂₉×#8732, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 559);