C₃₇×#2040 #17990858

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2040

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₃₇×#537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

290 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 57
transitive degree	μ_t(G) = 1184

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

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#7416, C₃₇×#12141, C₃₇×#12143, …
maximal normal subgroup of	C₃₇×#7416, C₃₇×#12141, C₃₇×#12143, …
maximal quotient of	C₃₇×#10403, C₃₇×#10404, C₃₇×#10407, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 772);