C₃₇×#1687 #17990505

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1687

Subgroups

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

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

normal subgroups only

This group has 198 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)(39,42)(40,43)(44,45)(46,47)(48,50)(49,52)(51,53);(38,39,43,45,41,44,40,42);(39,44)(42,45)(47,50)(48,53)(51,52)

Conjugacy classes & character table

851 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12324, C₃₇×#12325, C₃₇×#12348, …
maximal normal subgroup of	C₃₇×#12324, C₃₇×#12325, C₃₇×#12348, …
maximal quotient of	C₃₇×#8652, C₃₇×#9229, C₃₇×#9571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 419);