C₃₇×#2313 #17991131

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₇×#2313

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂⁴×C₃₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

512 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 280 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) = 2368

A minimal faithful permutation representation — 4 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)(38,39,40,41)(42,43,46,44)(45,48,49,47)(52,56)(54,57);(43,44)(47,48)(50,51,53,55)(52,54,56,57);(42,44,46,43)(45,48,49,47)(50,52)(51,54)(53,56)(55,57);(42,45)(43,47)(44,48)(46,49)

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8737, C₃₇×#8755, C₃₇×#8758, …
maximal normal subgroup of	C₃₇×#8737, C₃₇×#8755, C₃₇×#8758, …
maximal quotient of	C₃₇×#13776, C₃₇×#13778, C₃₇×#13787, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1045);