C₅×#10219 #11932939

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#10219

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

564 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 29 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3207);