C₃₇×#1604 #17990422

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

Aliases

direct product

C₃₇×#1604

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₃₇×#537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

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

normal subgroups only

This group has 282 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) = 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)(38,39,41,44)(40,42,45,43)(50,53);(39,42)(40,45)(43,44)(46,47)(48,50)(49,51)(52,53);(46,48,49,52)(47,50,51,53)

Conjugacy classes & character table

851 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12126, C₃₇×#12128, C₃₇×#12129, …
maximal normal subgroup of	C₃₇×#12126, C₃₇×#12128, C₃₇×#12129, …
maximal quotient of	C₃₇×#7752, C₃₇×#7754, C₃₇×#7764, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 336);