C₃₇×#1614 #17990432

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1614

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₂⁴×C₃₇	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

236 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 68 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)(46,47,50,53)(48,54,56,61)(49,57,59,52)(51,58,60,55);(38,39,41,43)(40,45,44,42)(46,48,50,56)(47,51,53,60)(49,58,59,55)(52,61,57,54);(40,44)(42,45)

Conjugacy classes & character table

851 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12214, C₃₇×#12216, C₃₇×#12218, …
maximal normal subgroup of	C₃₇×#12214, C₃₇×#12216, C₃₇×#12218, …
maximal quotient of	C₃₇×#7789, C₃₇×#7797, C₃₇×#7802, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 346);