C₆₁×D₃₇ #17985253

order	4514
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₆₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₆₁×D₃₇
unambiguous semidirect product	C₃₇⋊(C₂×C₆₁)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₆₁	D₃₇
commutator, cosocle, hypocenter	C₃₇	C₂×C₆₁
Frattini	C₁	G
Fitting, socle	C₃₇×C₆₁	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₆₁

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);(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₆₁×D₃₇, C₆₁×D₁₁₁, D₃₇×C₆₁⋊C₃, …
maximal normal subgroup of	C₃×C₆₁×D₃₇, D₃₇×C₆₁⋊C₃, C₆₁×D₃₇⋊C₃, …
maximal quotient of	C₃×C₆₁×D₃₇, C₆₁×D₁₁₁, C₅×C₆₁×D₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4514, 1);

C61×D37 #17985253