C₃₇×#2165 #17990983

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃₇×#2165

Subgroups

	subgroup	G/H
center	C₄×C₃₇	#117
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
socle	C₂×C₃₇	#432
prime radical	C₂³×C₃₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ #97×C₃₇ ◁ G

128 conjugacy classes of subgroups.

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)(38,39,40,45,41,46,48,53,42,43,47,50,49,44,51,52);(39,43)(40,47)(44,46)(48,51);(39,44,43,46)(40,47)(45,53,50,52)(48,51)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7225, C₃₇×#7226, C₃₇×#11521, …
maximal normal subgroup of	C₃₇×#7225, C₃₇×#7226, C₃₇×#11521, …
maximal quotient of	C₃₇×#12595, C₃₇×#12607, C₃₇×#12659, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 897);

C37×#2165 #17990983