C₃₇×#2080 #17990898

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2080

Subgroups

	subgroup	G/H
center	C₄×C₃₇	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂×C₃₇	#397
prime radical	C₂×#41×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 72 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)(40,48,51,61)(41,45)(42,46)(43,47)(44,50,58,55)(49,64,52,63)(53,59)(54,60)(56,62)(57,66,68,65)(67,69);(38,40)(39,44)(41,52)(42,55)(43,51)(45,57)(46,61)(47,58)(48,62)(49,54)(50,56)(53,65)(59,64)(60,68)(63,69)(66,67);(38,41,42,53,43,54,56,67)(39,45,46,59,47,60,62,69)(40,49,50,65,51,52,55,66)(44,57,48,63,58,68,61,64)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#2080, C₃₇×#23649, C₃₇×#23833, …
maximal normal subgroup of	C₂×C₃₇×#2080, C₃₇×#23649, C₃₇×#23833, …
maximal quotient of	C₃₇×#10627, C₃₇×#10628, C₃₇×#10629, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 812);