C₃₇×#2190 #17991008

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#2190

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	#447
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂×C₄×C₃₇ ◁ C₂×M₁₆×C₃₇ ◁ G

226 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 53
transitive degree	μ_t(G) = 592

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

Conjugacy classes & character table

740 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12114, C₃₇×#12116, C₃₇×#12117, …
maximal normal subgroup of	C₃₇×#12114, C₃₇×#12116, C₃₇×#12117, …
maximal quotient of	C₃₇×#12102, C₃₇×#12103, C₃₇×#12104, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 922);