C₃₇×#2053 #17990871

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2053

Subgroups

	subgroup	G/H
center	C₄×C₃₇	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂×C₃₇	#394
prime radical	C₃₇×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

264 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 80 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)(40,49)(41,54)(42,59)(43,47)(44,48)(45,51)(46,56)(50,60)(52,62)(53,61)(55,63)(57,64)(58,67)(65,68)(66,69);(38,40)(39,45)(41,55)(42,52)(43,51)(44,53)(46,63)(47,61)(48,62)(49,59)(50,57)(54,60)(56,66)(58,65)(64,68)(67,69);(38,41,42,56,44,58,43,57)(39,46,47,54,48,64,59,67)(40,50,51,65,53,66,52,55)(45,60,49,63,62,69,61,68)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12170, C₃₇×#12171, C₃₇×#12172, …
maximal normal subgroup of	C₃₇×#12170, C₃₇×#12171, C₃₇×#12172, …
maximal quotient of	C₃₇×#10464, C₃₇×#10465, C₃₇×#10466, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 785);