C₅₃×D₃₇ #16840969

order	3922
isoclinism	isoclinic class of D₃₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₃×D₃₇
unambiguous semidirect product	C₃₇⋊(C₂×C₅₃)

Splittings

Fitting, socle, Hall	C₃₇×C₅₃ ⋊ C₂
derived, Sylow	C₃₇ ⋊ C₂×C₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₃₇
commutator, cosocle, hypocenter	C₃₇	C₂×C₅₃
Frattini	C₁	G
Fitting, socle	C₃₇×C₅₃	C₂
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₅₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 1961

A minimal faithful permutation representation — 2 generators on 90 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);(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Conjugacy classes & character table

1060 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅₃×D₃₇, C₅₃×D₁₁₁, C₅₃×D₃₇⋊C₃, …
maximal normal subgroup of	C₃×C₅₃×D₃₇, C₅₃×D₃₇⋊C₃, C₅×C₅₃×D₃₇, …
maximal quotient of	C₃×C₅₃×D₃₇, C₅₃×D₁₁₁, C₅×C₅₃×D₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3922, 1);

C53×D37 #16840969