C₅×#10338 #11933058

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10336
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#10338

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂⁵	C₂³×C₅
prime radical	C₂³×C₅×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂⁵×C₅ ◁ G

998 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 29 points:

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

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, 3326);