C₅×#2646 #10616241

order	640
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#2646

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

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

normal subgroups only

This group has 436 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)(6,7,10,11)(8,9,12,13)(15,16)(19,20)(22,23)(24,27)(25,28)(26,29);(7,11)(8,12)(14,15,18,16)(17,19,21,20)(22,24)(23,26)(25,29)(27,28);(6,8)(7,9)(10,12)(11,13)(14,16,18,15)(17,19,21,20)(22,24)(23,27)(25,29)(26,28);(6,9)(7,12)(8,11)(10,13)(14,17)(15,19)(16,20)(18,21)(22,23)(24,26)(25,28)(27,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12002287, #12002292, #12002297, …
maximal subgroup of	C₅×#10344, C₅×#10345, C₅×#10346, …
maximal normal subgroup of	C₅×#10344, C₅×#10345, C₅×#10346, …
maximal quotient of	C₅×#14265, C₅×#14275, C₅×#14329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18323);