C₅×#464 #65627

order	320
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#464

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅
prime radical	C₂×C₅×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

112 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10612299, #10612300, #10612301, …
maximal subgroup of	#10612299, #10612300, #10612301, …
maximal normal subgroup of	#10612299, #10612300, #10612301, …
maximal quotient of	C₅×#1678, C₅×#1680, C₅×#1682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 965);