C₇×C₁₉⋊C₃×A₅ #37561654

order	23940
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

direct product	C₇×C₁₉⋊C₃×A₅
unambiguous semidirect product	C₁₉⋊(C₃×C₇×A₅)
unambiguous semidirect product	(C₁₉×A₅)⋊(C₃×C₇)

Splittings

Fitting, Hall	C₇×C₁₉ ⋊ C₃×A₅
socle	C₇×C₁₉×A₅ ⋊ C₃
derived	C₁₉×A₅ ⋊ C₃×C₇
Sylow	C₁₉ ⋊ C₃×C₇×A₅

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₁₉⋊C₃×A₅
commutator, hypocenter	C₁₉×A₅	C₃×C₇
Frattini	C₁	G
Fitting	C₇×C₁₉	C₃×A₅
socle	C₇×C₁₉×A₅	C₃
cosocle	C₁₉	C₃×C₇×A₅
LCS	G ▷ C₁₉×A₅
UCS	C₁ ◁ C₇

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 665

A minimal faithful permutation representation — 2 generators on 31 points:

(8,9,11)(10,13,17)(12,16,18)(14,19,22)(15,20,24)(21,23,26)(27,31,30,29,28);(1,2,3,4,5,6,7)(8,10,14,9,12,11,15,20,22,25,13,18,16,21,24,26,17,19,23)(27,28,30,29,31)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 786);

C7×C19⋊C3×A5 #37561654