D₅×(C₁₃⋊C₉)⋊D₁₃ #44477694

order	30420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₅×(C₁₃⋊C₉)⋊D₁₃
unambiguous semidirect product	(D₅×C₁₃⋊C₉)⋊D₁₃

Splittings

derived, Hall	C₅×C₁₃² ⋊ C₂²×C₉
Hall	C₅×C₁₃×C₁₃⋊C₉ ⋊ C₂²
Hall	D₅×C₁₃⋊D₁₃ ⋊ C₉
Hall	C₁₃×C₁₃⋊C₉ ⋊ C₂×D₅
Sylow	C₁₃² ⋊ C₂×C₉×D₅
Sylow	C₅ ⋊ C₂×(C₁₃⋊C₉)⋊D₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₅×(C₁₃⋊C₃)⋊D₁₃
commutator, hypocenter	C₅×C₁₃²	C₂²×C₉
Fitting, socle, cosocle	C₃×C₅×C₁₃²	C₂²×C₃
prime radical	C₃×D₅×C₁₃⋊D₁₃	C₃
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₃

200 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 40
transitive degree	μ_t(G) = 585

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,4,5,3)(6,10,14,8,13,9,11,7,12)(16,20,21,17,18,22)(19,25,26,23,24,27)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40);(2,3)(4,5)(15,16,19,21,20,24,26,27,25,18,22,23,17)(28,29,31,33,35,37,39,40,38,36,34,32,30)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 283);