C₅×#21796 #11944516

order	1280
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₅×#21796

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂×C₅×#537	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₅×C₄⋊C₄ ◁ G

1334 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 366 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)(6,7)(8,12)(9,11)(10,13)(14,15)(16,22,24,19)(17,20)(18,21)(23,28,29,26)(25,27);(6,8,9,13)(7,10,11,12)(17,25)(20,27)(23,29)(26,28);(8,13)(10,12)(14,16)(15,19)(17,23)(18,24)(20,26)(21,22)(25,29)(27,28);(7,11)(10,12)(14,17,18,25)(15,20,21,27)(16,23,24,29)(19,26,22,28)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 14784);