D₅×#72209831 #72253740

order	15360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

D₅×#72209831

Splittings

Fitting	C₅×#6499874 ⋊ C₂×C₃
derived	C₅×#62817 ⋊ C₂²×C₃
Hall	D₅×#6499874 ⋊ C₃
Sylow	C₅ ⋊ C₂×#72209831

Subgroups

	subgroup	G/H
center	C₂	D₅×#35346⋊C₃
commutator, cosocle, hypocenter	C₅×#62817	C₂²×C₃
Frattini	C₂⁴	C₂×D₅×C₂²⋊A₄
Fitting	C₅×#6499874	C₂×C₃
socle	C₂×C₅	C₂×#35346⋊C₃
hypercenter	C₂²	D₅×#85645
LCS	G ▷ C₅×#62817
UCS	C₁ ◁ C₂ ◁ C₂²

At least 12852 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has at least 12852 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(2,3)(4,5)(8,9,14)(11,12,17)(15,16,20)(18,19,21);(1,2,4,5,3)(6,7,10,13)(8,18)(9,19)(11,15)(12,16)(14,17,20,21);(6,14)(7,17)(8,9)(10,20)(11,12)(13,21)(15,16)(18,19)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3)(4,5)(8,9,14)(11,12,17)(15,16,20)(18,19,21), (1,2,4,5,3)(6,7,10,13)(8,18)(9,19)(11,15)(12,16)(14,17,20,21), (6,14)(7,17)(8,9)(10,20)(11,12)(13,21)(15,16)(18,19));