C₃₇×#2214 #17991032

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2214

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₇
prime radical	#135×C₃₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂×C₄×C₃₇ ◁ C₄×C₈×C₃₇ ◁ G

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 1184

A minimal faithful permutation representation — 3 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)(38,39,45,51)(40,49,54,60)(41,55,57,68)(42,50,44,48)(43,46,53,61)(47,62,64,52)(56,69,59,65)(58,66,67,63);(38,40,44,53,45,54,42,43)(39,46,50,60,51,61,48,49)(41,52,59,67,57,62,56,58)(47,55,66,69,64,68,63,65);(38,41)(39,47)(40,52)(42,56)(43,58)(44,59)(45,57)(46,55)(48,63)(49,65)(50,66)(51,64)(53,67)(54,62)(60,69)(61,68)

Conjugacy classes & character table

851 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12312, C₃₇×#12313, C₃₇×#12315, …
maximal normal subgroup of	C₃₇×#12312, C₃₇×#12313, C₃₇×#12315, …
maximal quotient of	C₃₇×#12422, C₃₇×#12423, C₃₇×#12426, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 946);