C₃₇×#1955 #17990773

order	4736
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#1955

Subgroups

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

384 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 2368

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7060, C₃₇×#7105, C₃₇×#7283, …
maximal normal subgroup of	C₃₇×#7060, C₃₇×#7105, C₃₇×#7283, …
maximal quotient of	C₃₇×#9643, C₃₇×#9644, C₃₇×#9645, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 687);