C₂₉×#1659 #16815373

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₂₉×#1659

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂₉×H(3,𝔽₄)	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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)(38,39,43,47)(40,48,50,53)(41,46,42,45)(44,51,52,49);(32,36)(34,37)(38,40)(39,44)(41,51)(42,49)(43,50)(45,53)(46,48)(47,52);(30,31)(32,37)(33,35)(34,36)(38,40)(39,44)(41,49)(42,51)(43,50)(45,48)(46,53)(47,52)

Conjugacy classes & character table

580 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₉×#13678, C₂×C₂₉×#1659, C₂₉×#18984, …
maximal normal subgroup of	C₂₉×#13678, C₂×C₂₉×#1659, C₂₉×#18984, …
maximal quotient of	C₂₉×#7750, C₂₉×#7755, C₂₉×#7758, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 391);