D₅×#72209641 #72246371

order	15360
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

D₅×#72209641

Splittings

Fitting	C₅×#62817 ⋊ C₂×D₃
derived	C₅×#11710730 ⋊ C₂²
Sylow	C₅ ⋊ C₂×#72209641

Subgroups

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

40954 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 40954 conjugacy classes of subgroups — too many to draw. Only its 85 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)(7,12,13)(8,10,11)(15,16,18)(19,20,21);(1,2,4,5,3)(7,10)(11,13)(15,18,19,21)(16,20);(6,10)(7,8)(9,13)(11,12)(14,18)(15,16)(17,21)(19,20)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#72209641
maximal normal subgroup of	AGL(1,𝔽₅)×#72209641

GAP

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

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