C₃₇×#3305 #17992123

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3305

Subgroups

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

408 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)(38,39,41,43)(40,44,45,42)(46,47,51,54)(48,52,50,58)(49,56,57,60)(53,59,61,55);(38,40)(39,42)(41,45)(43,44)(46,48,51,50)(47,52,54,58)(49,55,57,59)(53,60,61,56);(46,49,51,57)(47,53,54,61)(48,55,50,59)(52,60,58,56);(40,45)(42,44)(46,48,51,50)(47,52,54,58)(49,55,57,59)(53,60,61,56)

Conjugacy classes & character table

1295 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#19783, C₃₇×#19784, C₃₇×#19792, …
maximal normal subgroup of	C₃₇×#19783, C₃₇×#19784, C₃₇×#19792, …
maximal quotient of	C₃₇×#22418, C₃₇×#22489, C₃₇×#22573, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2037);

C37×#3305 #17992123