C₅×#23469 #11946189

order	1280
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₅×#23469

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

1402 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 334 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) = 160

A minimal faithful permutation representation — 4 generators on 29 points:

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

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, 16457);