C₃₇×#1641 #17990459

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

Aliases

direct product

C₃₇×#1641

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	C₃₇×#550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ 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) = 61
transitive degree	μ_t(G) = 2368

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)(38,39)(40,42)(41,44)(43,45)(46,47,50,53)(48,54,55,61)(49,57,58,52)(51,59,60,56);(39,42)(40,45)(43,44)(49,58)(52,57)(54,61)(56,59);(46,48,50,55)(47,51,53,60)(49,59,58,56)(52,61,57,54)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#1641, C₃₇×#18515, C₃₇×#18516, …
maximal normal subgroup of	C₂×C₃₇×#1641, C₃₇×#18515, C₃₇×#18516, …
maximal quotient of	C₃₇×#7756, C₃₇×#7758, C₃₇×#7762, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 373);