C₅×D₁₃×D₁₁₇ #44477749

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

Aliases

direct product	C₅×D₁₃×D₁₁₇
unambiguous semidirect product	(C₉×D₁₃)⋊(C₅×D₁₃)
unambiguous semidirect product	(C₁₃×D₁₃)⋊(C₅×D₉)

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

192 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

2400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 338);