C₃₇×#2986 #17991804

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#2986

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂³×C₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ G

330 conjugacy classes of subgroups; only the 256 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 256 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) = 69
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 69 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)(40,46)(41,44)(42,45)(43,48)(47,52)(49,51)(50,53)(54,55,57,60,58,61,64,67)(56,59,63,66,62,65,68,69);(38,40,42,48)(39,43,45,46)(41,49,50,47)(44,52,53,51)(54,56)(55,59)(57,63)(58,62)(60,66)(61,65)(64,68)(67,69);(38,41,42,50)(39,44,45,53)(40,47,48,49)(43,51,46,52)(56,62)(59,65)(63,68)(66,69);(39,45)(43,46)(44,53)(51,52)(56,62)(59,65)(63,68)(66,69)

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8936, C₃₇×#9068, C₃₇×#9070, …
maximal normal subgroup of	C₃₇×#8936, C₃₇×#9068, C₃₇×#9070, …
maximal quotient of	C₃₇×#20538, C₃₇×#20540, C₃₇×#20571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1718);