C₃₇×#2858 #17991676

order	4736
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₃₇×#2858

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

752 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 264 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 — 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)(39,40)(43,44)(48,51)(49,53);(38,39,41,40)(42,43,45,44)(46,47,50,52)(48,49,51,53);(42,43,45,44)(46,48,50,51)(47,49,52,53);(38,40,41,39)(42,44,45,43)(46,49)(47,51)(48,52)(50,53)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7659, C₃₇×#7679, C₃₇×#7699, …
maximal normal subgroup of	C₃₇×#7659, C₃₇×#7679, C₃₇×#7699, …
maximal quotient of	C₃₇×#14687, C₃₇×#14700, C₃₇×#14708, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1590);