C₃₇×#3414 #17992232

order	4736
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₇×#3414

Subgroups

	subgroup	G/H
center	C₄×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃₇
socle	C₂×C₃₇	C₂²×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 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)(41,47)(42,54)(43,58)(44,46)(45,51)(48,56)(50,66)(52,65)(53,63)(55,61)(57,68)(59,64)(60,67)(62,69);(38,40)(39,45)(41,50)(42,55)(43,59)(44,52)(46,60)(47,61)(48,64)(49,54)(51,56)(53,67)(57,69)(58,62)(63,66)(65,68);(39,46)(45,60)(47,58)(48,63)(49,65)(54,68)(61,62)(64,66);(38,41,44,43)(39,47,46,58)(40,50,52,59)(42,53,57,56)(45,61,60,62)(48,54,63,68)(49,66,65,64)(51,55,67,69)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#13021, C₃₇×#13024, C₃₇×#13027, …
maximal normal subgroup of	C₃₇×#13021, C₃₇×#13024, C₃₇×#13027, …
maximal quotient of	C₃₇×#33380, C₃₇×#33409, C₃₇×#33451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2146);