C₅×#575 #66239

order	320
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#575

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₅
socle	C₂×C₅	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ G

262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10602905, #10602906, #10602907, …
maximal subgroup of	#10602905, #10602906, #10602907, …
maximal normal subgroup of	#10602905, #10602906, #10602907, …
maximal quotient of	C₅×#2896, C₅×#2911, C₅×#2943, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1577);