C₃₇×#1535 #17990353

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#1535

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	#121×C₃₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

200 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 84 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)(38,39,42,47,43,46,41,45)(40,48,49,53,50,44,51,52)(54,55)(56,62,64,69)(57,61)(58,60)(59,63,67,65)(66,68);(38,40)(39,44)(41,49)(42,51)(43,50)(45,52)(46,48)(47,53)(54,56)(55,59)(57,65)(58,64)(60,67)(61,69)(62,68)(63,66);(38,41,43,42)(39,45,46,47)(40,49,50,51)(44,52,48,53)(55,60)(59,67)(61,68)(62,69)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#8489, C₃₇×#8490, C₃₇×#8500, …
maximal normal subgroup of	C₃₇×#8489, C₃₇×#8490, C₃₇×#8500, …
maximal quotient of	C₃₇×#7594, C₃₇×#7596, C₃₇×#7599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 267);