C₅×#72209641 #72246135

order	7680
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#72209641

Splittings

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

Subgroups

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

5062 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 5062 conjugacy classes of subgroups — too many to draw. Only its 50 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)(7,15,19,14)(8,12,20,13)(11,21)(17,18);(7,8,11)(12,14,17)(13,15,18)(19,20,21);(6,7)(8,11)(9,13)(10,12)(14,17)(15,18)(16,19)(20,21)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#72209641
maximal normal subgroup of	D₅×#72209641

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