D₅×#72209772 #72258497

order	15360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

D₅×#72209772

Splittings

Fitting	C₅×#62817 ⋊ C₂×D₃
derived	C₅×#11710730 ⋊ C₂²
Sylow	C₅ ⋊ C₂×#72209772

Subgroups

	subgroup	G/H
center	C₂	D₅×#11715850
commutator, cosocle, hypocenter	C₅×#11710730	C₂²
Frattini, hypercenter	C₂²	D₅×#87588
Fitting	C₅×#62817	C₂×D₃
socle	C₂×C₅	C₂×#11715850
LCS	G ▷ C₅×#11710730
UCS	C₁ ◁ C₂ ◁ C₂²

At least 10794 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has at least 10794 conjugacy classes of subgroups — too many to draw. Only its 67 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)(8,16,14)(9,20,15)(11,19,18)(12,21,17);(1,2,4,5,3)(6,11,20,19)(7,15,21,9)(8,18,16,13)(10,17,14,12);(8,15)(11,17)(12,19)(14,20)

Conjugacy classes & character table

132 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)(8,16,14)(9,20,15)(11,19,18)(12,21,17), (1,2,4,5,3)(6,11,20,19)(7,15,21,9)(8,18,16,13)(10,17,14,12), (8,15)(11,17)(12,19)(14,20));