C₃₇×#1989 #17990807

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1989

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

260 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 57
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 57 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)(39,41)(42,43)(44,48)(45,46)(47,49)(50,51,52,56,53,55,57,54);(38,39,40,41)(43,46)(47,48)(51,54)(52,57)(55,56);(42,44,45,49)(43,47,46,48)(51,55)(54,56)

Conjugacy classes & character table

1406 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8512, C₃₇×#8513, C₃₇×#10509, …
maximal normal subgroup of	C₃₇×#8512, C₃₇×#8513, C₃₇×#10509, …
maximal quotient of	C₃₇×#9866, C₃₇×#9867, C₃₇×#9900, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 721);