C₃₇×#1701 #17990519

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃₇×#1701

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	C₃₇×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ G

176 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 64 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)(39,42)(40,43)(44,45)(46,47)(48,50)(49,51)(52,53);(38,39,43,45,41,44,40,42)(47,50)(51,52);(39,44)(42,45)(47,51)(49,53)(50,52)

Conjugacy classes & character table

851 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#1701, C₃₇×#19142, C₃₇×#19157, …
maximal normal subgroup of	C₂×C₃₇×#1701, C₃₇×#19142, C₃₇×#19157, …
maximal quotient of	C₃₇×#8656, C₃₇×#9225, C₃₇×#9229, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 433);