C₃₇×#2050 #17990868

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2050

Subgroups

	subgroup	G/H
center	C₄×C₃₇	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂×C₃₇	#394
prime radical	C₃₇×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

266 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 592

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

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#7414, C₃₇×#7415, C₃₇×#12173, …
maximal normal subgroup of	C₃₇×#7414, C₃₇×#7415, C₃₇×#12173, …
maximal quotient of	C₃₇×#10456, C₃₇×#10457, C₃₇×#10458, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 782);