C₃₇×#3026 #17991844

order	4736
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₇×#3026

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₇
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂⁴×C₃₇ ◁ G

800 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 212 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) = 592

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

Conjugacy classes & character table

1073 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#23895, C₃₇×#23896, C₃₇×#23996, …
maximal normal subgroup of	C₃₇×#23895, C₃₇×#23896, C₃₇×#23996, …
maximal quotient of	C₃₇×#18284, C₃₇×#18285, C₃₇×#18286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1758);