C₅×#2607 #10616202

order	640
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#2607

Subgroups

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

444 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 29 points:

(1,2,3,4,5)(14,15)(16,20)(17,18)(19,21)(22,23,26,25)(24,28,29,27);(7,10)(11,13)(15,18)(19,20)(24,29)(27,28);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,18)(20,21)(22,24,26,29)(23,27,25,28);(6,8,9,12)(7,11,10,13)(14,16)(15,19)(17,21)(18,20)(22,25,26,23)(24,28,29,27)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12002529, #12002534, #12011506, …
maximal subgroup of	C₅×#10386, C₅×#10387, C₅×#12121, …
maximal normal subgroup of	C₅×#10386, C₅×#10387, C₅×#12121, …
maximal quotient of	C₅×#13953, C₅×#14091, C₅×#14093, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18284);