C₅×D₃×C₃⋊(C₁₃⋊D₁₃) #44478275

order	30420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₁₃

Aliases

direct product	C₅×D₃×C₃⋊(C₁₃⋊D₁₃)
unambiguous semidirect product	(C₃×D₃)⋊(C₅×C₁₃⋊D₁₃)
unambiguous semidirect product	(D₃×C₁₃)⋊(C₅×D₃₉)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₃² ⋊ C₂²
derived, Hall	C₃²×C₁₃² ⋊ C₂²×C₅
Hall	C₅×C₁₃² ⋊ D₃²
Sylow	C₁₃² ⋊ C₅×D₃²
Hall	C₃²×C₅ ⋊ C₂×C₁₃⋊D₁₃
Sylow	C₃² ⋊ C₂×C₅×C₁₃⋊D₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₃×C₃⋊(C₁₃⋊D₁₃)
commutator, cosocle, hypocenter	C₃²×C₁₃²	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₃²	C₂²
LCS	G ▷ C₃²×C₁₃²
UCS	C₁ ◁ C₅

704 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 5070

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

3825 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 864);