C₃₇×#2816 #17991634

order	4736
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₇×#2816

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂×#121×C₃₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

528 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 184 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)(43,46)(47,49)(50,51,54,57)(52,53,56,55);(39,41)(43,46)(44,48)(50,52,54,56)(51,55,57,53);(38,39)(40,41)(42,43)(44,47)(45,46)(48,49)(50,52)(51,55)(53,57)(54,56);(42,44)(43,47)(45,48)(46,49)(50,53,54,55)(51,56,57,52)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#9981, C₃₇×#9984, C₃₇×#18451, …
maximal normal subgroup of	C₃₇×#9981, C₃₇×#9984, C₃₇×#18451, …
maximal quotient of	C₃₇×#14622, C₃₇×#14624, C₃₇×#14625, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1548);