C₂₉×#1644 #16815358

order	3712
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₉×#1644

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂×D₄×C₂₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 1856

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

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1644, C₂₉×#18485, C₂₉×#18519, …
maximal normal subgroup of	C₂×C₂₉×#1644, C₂₉×#18485, C₂₉×#18519, …
maximal quotient of	C₂₉×#7774, C₂₉×#7777, C₂₉×#7782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 376);