C₃₇×#1532 #17990350

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1532

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	#121×C₃₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 61
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 3 generators on 61 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)(39,43,44,45)(46,47,50,53,51,54,49,58)(48,55,56,60,57,52,59,61);(38,39)(40,43)(41,45)(42,44)(46,48)(47,52)(49,56)(50,59)(51,57)(53,60)(54,55)(58,61);(38,40,42,41)(39,43,44,45)(46,49,51,50)(47,53,54,58)(48,56,57,59)(52,60,55,61)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8348, C₃₇×#8349, C₃₇×#8497, …
maximal normal subgroup of	C₃₇×#8348, C₃₇×#8349, C₃₇×#8497, …
maximal quotient of	C₃₇×#7580, C₃₇×#7581, C₃₇×#7586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 264);