C₃₇×D₆₃ #17987948

order	4662
isoclinism	isoclinic class of D₆₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃₇×D₆₃
unambiguous semidirect product	C₇⋊(D₉×C₃₇)
unambiguous semidirect product	C₉⋊(D₇×C₃₇)

Splittings

Fitting, Hall	C₇×C₉×C₃₇ ⋊ C₂
derived, Hall	C₇×C₉ ⋊ C₂×C₃₇
Hall	C₉×C₃₇ ⋊ D₇
Hall	C₇×C₃₇ ⋊ D₉
Sylow	C₉ ⋊ D₇×C₃₇
Sylow	C₇ ⋊ D₉×C₃₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃₇	D₆₃
commutator, cosocle, hypocenter	C₇×C₉	C₂×C₃₇
Frattini	C₃	C₃₇×D₂₁
Fitting	C₇×C₉×C₃₇	C₂
socle	C₃×C₇×C₃₇	D₃
LCS	G ▷ C₇×C₉
UCS	C₁ ◁ C₃₇

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 2331

A minimal faithful permutation representation — 2 generators on 53 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);(39,46)(40,45)(41,44)(42,43)(48,53)(49,52)(50,51)

Conjugacy classes & character table

1221 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×C₃₇×D₆₃, C₃₇×D₃₁₅, C₃₇×D₄₄₁, …
maximal normal subgroup of	C₅×C₃₇×D₆₃, C₇×C₃₇×D₆₃
maximal quotient of	C₅×C₃₇×D₆₃, C₃₇×D₃₁₅, C₃₇×D₄₄₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4662, 23);

C37×D63 #17987948