C₅×#2383 #10615978

order	640
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#2383

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator	C₂²	C₂³×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

456 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 264 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) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 25 points:

(1,2,3,4,5)(6,7,8,9)(11,14)(12,16)(18,19,22,20)(21,24,25,23);(11,14)(15,17)(19,20)(21,25);(10,11)(12,15)(13,14)(16,17)(18,20,22,19)(21,24,25,23);(10,12,13,16)(11,15,14,17)(18,21)(19,23)(20,24)(22,25)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₅×#2383, #12261969, #12261987, …
maximal subgroup of	C₂×C₅×#2383, C₅×#34350, C₄×C₅×#560, …
maximal normal subgroup of	C₂×C₅×#2383, C₅×#34350, C₄×C₅×#560, …
maximal quotient of	C₅×#13830, C₅×#13831, C₅×#13833, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18060);