C₃₇×#2042 #17990860

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2042

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

318 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 100 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)(38,39)(40,41)(43,44)(46,47,51,53)(48,49,52,50);(42,43)(44,45)(46,48)(47,49)(50,53)(51,52);(39,41)(42,44,45,43)(47,52)(48,53)(49,50)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×#2042, C₃₇×#23504, C₃₇×#23506, …
maximal normal subgroup of	C₂×C₃₇×#2042, C₃₇×#23504, C₃₇×#23506, …
maximal quotient of	C₃₇×#10404, C₃₇×#10405, C₃₇×#10410, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 774);