C₃₇×#1618 #17990436

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1618

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₃₇×#538	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

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

normal subgroups only

This group has 240 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) = 61
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 generators on 61 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)(40,44)(43,45)(46,47,51,54)(48,55,57,60)(49,53,50,58)(52,59,61,56);(38,39)(40,45)(41,43)(42,44)(46,48)(47,52)(49,56)(50,59)(51,57)(53,60)(54,61)(55,58);(40,44)(42,45)(46,49,51,50)(47,53,54,58)(48,56,57,59)(52,60,61,55)

Conjugacy classes & character table

851 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#1618, C₃₇×#18121, C₃₇×#18127, …
maximal normal subgroup of	C₂×C₃₇×#1618, C₃₇×#18121, C₃₇×#18127, …
maximal quotient of	C₃₇×#7790, C₃₇×#7793, C₃₇×#7796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 350);