C₃₇×#3177 #17991995

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

Aliases

direct product

C₃₇×#3177

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ #117×C₃₇ ◁ G

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 77 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)(46,47,51,55)(48,52,57,65)(49,59,62,74)(50,63,60,54)(53,67,68,61)(56,69,71,77)(58,73,70,66)(64,75,76,72);(39,42)(43,45)(46,48,51,57)(47,52,55,65)(49,56,62,71)(50,58,60,70)(53,64,68,76)(54,66,63,73)(59,69,74,77)(61,72,67,75);(46,49,60,67,51,62,50,61)(47,53,63,74,55,68,54,59)(48,56,70,75,57,71,58,72)(52,64,73,77,65,76,66,69);(38,40,41,44)(39,43,42,45)(48,57)(52,65)(56,71)(58,70)(64,76)(66,73)(69,77)(72,75)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#11771, C₃₇×#11914, C₃₇×#11990, …
maximal normal subgroup of	C₃₇×#11771, C₃₇×#11914, C₃₇×#11990, …
maximal quotient of	C₃₇×#20821, C₃₇×#21281, C₃₇×#21289, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1909);

C37×#3177 #17991995