C₃₇×#2052 #17990870

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2052

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₂×#41×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

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

normal subgroups only

This group has 262 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) = 2368

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

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#2052, C₃₇×#23530, C₃₇×#23532, …
maximal normal subgroup of	C₂×C₃₇×#2052, C₃₇×#23530, C₃₇×#23532, …
maximal quotient of	C₃₇×#10458, C₃₇×#10459, C₃₇×#10462, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 784);