C₅×#2644 #10616239

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

Aliases

direct product

C₅×#2644

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

420 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 184 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)(7,10)(11,12)(14,15)(16,19)(17,20)(18,21)(22,23,26,28)(24,29,25,27);(6,7,9,10)(8,11,13,12)(14,16)(15,17)(18,20)(19,21)(22,24,26,25)(23,27,28,29);(6,7)(8,12)(9,10)(11,13)(14,16)(15,19)(17,21)(18,20)(22,23)(24,27)(25,29)(26,28);(6,8,9,13)(7,11,10,12)(14,17,18,19)(15,20,21,16)(22,25,26,24)(23,27,28,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12202441, #12202456, #12202531, …
maximal subgroup of	C₅×#29436, C₅×#29437, C₅×#29442, …
maximal normal subgroup of	C₅×#29436, C₅×#29437, C₅×#29442, …
maximal quotient of	C₅×#14260, C₅×#14266, C₅×#14331, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18321);