C₃₇×#1659 #17990477

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1659

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	C₃₇×H(3,𝔽₄)	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ G

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 1184

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

Conjugacy classes & character table

740 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#13678, C₂×C₃₇×#1659, C₃₇×#18984, …
maximal normal subgroup of	C₃₇×#13678, C₂×C₃₇×#1659, C₃₇×#18984, …
maximal quotient of	C₃₇×#7750, C₃₇×#7755, C₃₇×#7758, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 391);