C₅×D₁₁₇⋊D₁₃ #44477750

order	30420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₁₁₇⋊D₁₃
unambiguous semidirect product	D₁₁₇⋊(C₅×D₁₃)

Splittings

Fitting, Hall	C₅×C₉×C₁₃² ⋊ C₂²
derived, Hall	C₉×C₁₃² ⋊ C₂²×C₅
Hall	C₅×C₁₃² ⋊ C₂×D₉
Sylow	C₁₃² ⋊ C₂×C₅×D₉
Hall	C₅×C₉ ⋊ D₁₃²
Sylow	C₉ ⋊ C₅×D₁₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₁₇⋊D₁₃
commutator, cosocle, hypocenter	C₉×C₁₃²	C₂²×C₅
Frattini	C₃	C₅×D₃₉⋊D₁₃
Fitting	C₅×C₉×C₁₃²	C₂²
socle	C₃×C₅×C₁₃²	C₂×D₃
LCS	G ▷ C₉×C₁₃²
UCS	C₁ ◁ C₅

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1170

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,4,5)(6,11,9,8,13,10,12,7,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40);(7,9)(8,12)(10,13)(11,14)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29,31,33,35,37,39,40,38,36,34,32,30)

Conjugacy classes & character table

2040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 339);

C5×D117⋊D13 #44477750