C₃₇×#1628 #17990446

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1628

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	C₃₇×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ G

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

normal subgroups only

This group has 216 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) = 2368

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(39,43)(40,41)(44,45)(47,48)(49,50)(52,53);(38,39)(40,43)(41,45)(42,44)(46,47,51,53)(48,49,52,50);(38,40,42,41)(39,43,44,45)(46,48)(47,49)(50,53)(51,52)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#1628, C₃₇×#18494, C₃₇×#18520, …
maximal normal subgroup of	C₂×C₃₇×#1628, C₃₇×#18494, C₃₇×#18520, …
maximal quotient of	C₃₇×#7740, C₃₇×#7744, C₃₇×#7746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 360);