C₅×#20258913 #44477998

order	30420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₅×#20258913

Splittings

Fitting, socle	C₃×C₅×C₁₃² ⋊ C₂×D₃
derived, Hall	C₃×#92789 ⋊ C₂²×C₅
Hall	C₃×C₅×#92789 ⋊ C₂²
Hall	C₅×C₁₃² ⋊ C₂×C₃⋊D₃
Sylow	C₁₃² ⋊ C₂×C₅×C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#20258913
commutator, cosocle, hypocenter	C₃×#92789	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃²	C₂×D₃
LCS	G ▷ C₃×#92789
UCS	C₁ ◁ C₅

192 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 34 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) = 34
transitive degree	μ_t(G) = 390

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 587);