C₃×C₅×D₃₉⋊D₁₃ #44478238

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

Aliases

direct product

C₃×C₅×D₃₉⋊D₁₃

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₃² ⋊ C₂²
derived	C₃×C₁₃² ⋊ C₂²×C₃×C₅
Hall	C₃²×C₁₃² ⋊ C₂²×C₅
Hall	C₅×C₁₃² ⋊ C₂×C₃×D₃
Sylow	C₁₃² ⋊ C₂×C₃×C₅×D₃
Hall	C₃²×C₅ ⋊ D₁₃²
Sylow	C₃² ⋊ C₅×D₁₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	D₃₉⋊D₁₃
commutator, cosocle, hypocenter	C₃×C₁₃²	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃²	C₂²
LCS	G ▷ C₃×C₁₃²
UCS	C₁ ◁ C₃×C₅

320 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 37
transitive degree	μ_t(G) = 390

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

2250 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 827);