C₃₇×#2062 #17990880

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2062

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₃₇
prime radical	C₂⁴×C₃₇	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂⁴×C₃₇ ◁ G

314 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 1184

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

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12226, C₃₇×#12227, C₃₇×#12228, …
maximal normal subgroup of	C₃₇×#12226, C₃₇×#12227, C₃₇×#12228, …
maximal quotient of	C₃₇×#10567, C₃₇×#10568, C₃₇×#10569, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 794);