C₃₇×#3178 #17991996

order	4736
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3178

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	#121×C₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ #117×C₃₇ ◁ G

350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 61 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)(40,43)(41,42)(44,45)(48,54)(49,57)(51,58)(53,59)(55,60)(56,61);(39,42)(43,45)(46,47)(48,55)(49,53)(50,52)(51,56)(54,60)(57,59)(58,61);(46,48,49,54,50,56,57,61)(47,51,53,58,52,55,59,60);(38,40,41,44)(39,43,42,45)(47,52)(51,55)(53,59)(58,60)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₇×#3178, C₃₇×#60685, C₃₇×#60694, …
maximal normal subgroup of	C₂×C₃₇×#3178, C₃₇×#60685, C₃₇×#60694, …
maximal quotient of	C₃₇×#20822, C₃₇×#21283, C₃₇×#21285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1910);

C37×#3178 #17991996