C₂₉×#1605 #16815319

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

Aliases

direct product

C₂₉×#1605

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₉
prime radical	C₂³×C₂₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

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

normal subgroups only

This group has 238 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) = 53
transitive degree	μ_t(G) = 928

A minimal faithful permutation representation — 3 generators on 53 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)(34,37)(38,39,42,45)(40,46,48,53)(41,49,51,44)(43,50,52,47);(30,31)(32,34)(33,35)(36,37)(38,40,42,48)(39,43,45,52)(41,50,51,47)(44,53,49,46);(30,32,33,36)(31,34,35,37)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12220, C₂₉×#12222, C₂₉×#12223, …
maximal normal subgroup of	C₂₉×#12220, C₂₉×#12222, C₂₉×#12223, …
maximal quotient of	C₂₉×#7757, C₂₉×#7761, C₂₉×#7763, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 337);