C₂₉×#1611 #16815325

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1611

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

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

normal subgroups only

This group has 210 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)(31,34)(32,36)(35,37)(38,39,42,45)(40,43,47,50)(41,44,48,51)(46,52,53,49);(30,31,33,35)(32,37,36,34)(38,40)(39,43)(41,46)(42,47)(44,49)(45,50)(48,53)(51,52);(38,41,42,48)(39,44,45,51)(40,46,47,53)(43,49,50,52)

Conjugacy classes & character table

667 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#12209, C₂₉×#12210, C₂₉×#12211, …
maximal normal subgroup of	C₂₉×#12209, C₂₉×#12210, C₂₉×#12211, …
maximal quotient of	C₂₉×#7794, C₂₉×#7798, C₂₉×#7807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 343);