C₅×D₃×(C₁₃⋊C₃)⋊D₁₃ #44478002

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

Aliases

direct product	C₅×D₃×(C₁₃⋊C₃)⋊D₁₃
unambiguous semidirect product	(D₃×C₁₃⋊C₃)⋊(C₅×D₁₃)

Splittings

Fitting, socle	C₃×C₅×C₁₃² ⋊ C₂²×C₃
derived	C₃×C₁₃² ⋊ C₂²×C₃×C₅
Hall	C₃×C₅×C₁₃×C₁₃⋊C₃ ⋊ C₂²
Hall	C₃×C₁₃×C₁₃⋊C₃ ⋊ C₂²×C₅
Hall	C₅×C₁₃² ⋊ C₂×C₃×D₃
Sylow	C₁₃² ⋊ C₂×C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₃×(C₁₃⋊C₃)⋊D₁₃
commutator, cosocle, hypocenter	C₃×C₁₃²	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃²	C₂²×C₃
LCS	G ▷ C₃×C₁₃²
UCS	C₁ ◁ C₅

256 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 585

A minimal faithful permutation representation — 2 generators on 34 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 591);