C₃₇×#2309 #17991127

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₃₇×#2309

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂³×D₄×C₃₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

1104 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 360 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 — 4 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)(39,41)(42,43,44,45)(47,50)(48,52);(46,47,49,50)(48,51,52,53);(38,39)(40,41)(42,43)(44,45)(46,47)(48,51)(49,50)(52,53);(46,48)(47,51)(49,52)(50,53)

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8748, C₃₇×#8749, C₃₇×#9497, …
maximal normal subgroup of	C₃₇×#8748, C₃₇×#8749, C₃₇×#9497, …
maximal quotient of	C₃₇×#13775, C₃₇×#13785, C₃₇×#13795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1041);