C₃₇×#1664 #17990482

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1664

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	C₂×D₄×C₃₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ G

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

normal subgroups only

This group has 184 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) = 2368

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

Conjugacy classes & character table

740 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#1664, C₃₇×#18987, C₃₇×#18990, …
maximal normal subgroup of	C₂×C₃₇×#1664, C₃₇×#18987, C₃₇×#18990, …
maximal quotient of	C₃₇×#7764, C₃₇×#7765, C₃₇×#7766, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 396);