C₂₉×#1641 #16815355

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

Aliases

direct product

C₂₉×#1641

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₂₉
prime radical	C₂₉×#550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

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

normal subgroups only

This group has 250 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)(30,31)(32,34)(33,36)(35,37)(38,39,42,45)(40,46,47,53)(41,49,50,44)(43,51,52,48);(31,34)(32,37)(35,36)(41,50)(44,49)(46,53)(48,51);(38,40,42,47)(39,43,45,52)(41,51,50,48)(44,53,49,46)

Conjugacy classes & character table

754 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₂₉×#1641, C₂₉×#18515, C₂₉×#18516, …
maximal normal subgroup of	C₂×C₂₉×#1641, C₂₉×#18515, C₂₉×#18516, …
maximal quotient of	C₂₉×#7756, C₂₉×#7758, C₂₉×#7762, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3712, 373);