C₅×#7434 #11930154

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7434

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#353
commutator	C₂×C₄²	C₂×C₄×C₅
Frattini, cosocle	#390	C₂²×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#390 ◁ G

224 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 29 points:

(1,2,3,4,5)(7,10,11,12)(14,15)(16,23,25,19)(17,21,24,20)(18,22)(26,27,29,28);(6,7)(8,12)(9,11)(10,13)(14,16,24,29,18,26,17,25)(15,19,21,27,22,28,20,23)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 422);