C₅×D₃×#11759813 #44477995

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

Aliases

direct product

C₅×D₃×#11759813

Splittings

Fitting, socle	C₃×C₅×C₁₃² ⋊ C₂×D₃
derived, Hall	C₃×#92789 ⋊ C₂²×C₅
Hall	C₃×C₅×#92789 ⋊ C₂²
Hall	C₅×C₁₃² ⋊ D₃²
Sylow	C₁₃² ⋊ C₅×D₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₃×#11759813
commutator, cosocle, hypocenter	C₃×#92789	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃²	C₂×D₃
LCS	G ▷ C₃×#92789
UCS	C₁ ◁ C₅

154 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 34 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 584);