C₅×#2804 #10616399

order	640
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#2804

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

480 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 192 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)(8,12)(14,15,17,18)(16,19,20,21)(22,23)(24,27)(25,26)(28,29);(7,10)(11,13)(15,18)(16,20)(23,25)(24,28);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,18)(20,21)(22,24,26,28)(23,27,25,29);(6,8,9,12)(7,11,10,13)(14,16)(15,19)(17,20)(18,21)(22,25,26,23)(24,29,28,27)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11988058, #11988116, #11988296, …
maximal subgroup of	C₅×#7745, C₅×#7755, C₅×#7782, …
maximal normal subgroup of	C₅×#7745, C₅×#7755, C₅×#7782, …
maximal quotient of	C₅×#14440, C₅×#14457, C₅×#14471, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 18481);