C₅×#72209772 #72253864

order	7680
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#72209772

Splittings

Fitting	C₅×#62817 ⋊ D₃
derived	#11710730 ⋊ C₂×C₅

Subgroups

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

3174 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 3174 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) = 21
transitive degree	μ_t(G) = 80

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

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#72209772
maximal normal subgroup of	D₅×#72209772

GAP

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

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