C₃₇×D₆₁ #17985254

order	4514
isoclinism	isoclinic class of D₆₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×D₆₁
unambiguous semidirect product	C₆₁⋊(C₂×C₃₇)

Splittings

Fitting, socle, Hall	C₃₇×C₆₁ ⋊ C₂
derived, Sylow	C₆₁ ⋊ C₂×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃₇	D₆₁
commutator, cosocle, hypocenter	C₆₁	C₂×C₃₇
Frattini	C₁	G
Fitting, socle	C₃₇×C₆₁	C₂
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₃₇

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 2257

A minimal faithful permutation representation — 2 generators on 98 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98);(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₃₇×D₆₁, C₃₇×D₁₈₃, C₃₇×D₆₁⋊C₃, …
maximal normal subgroup of	C₃×C₃₇×D₆₁, C₃₇×D₆₁⋊C₃, C₃₇⋊C₃×D₆₁, …
maximal quotient of	C₃×C₃₇×D₆₁, C₃₇×D₁₈₃, C₅×C₃₇×D₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4514, 2);

C37×D61 #17985254