C₃×C₅×#11759813⋊C₂ #44477907

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

Aliases

direct product

C₃×C₅×#11759813⋊C₂

Splittings

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

Subgroups

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

160 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 32 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)(9,10,11,14,16,27,25,17,22,19,28,33,34,29,30,20,32,31,15,24,18,12,26,23,13,21);(11,22,25,13,15,18)(12,24,21,17,19,14)(16,32,34,26,28,30)(20,33,23,29,31,27)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 496);