C₅×#14349 #11937069

order	1280
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#14349

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

1638 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1638 conjugacy classes of subgroups — too many to draw. Only its 462 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) = 33
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 33 points:

(1,2,3,4,5)(10,11,14,12)(13,15,17,16)(19,22)(23,24)(26,27)(28,31)(29,32)(30,33);(11,12)(15,16)(18,19,21,22)(20,23,25,24)(26,28)(27,30)(29,33)(31,32);(6,7,8,9)(10,12,14,11)(13,15,17,16)(18,19)(20,24)(21,22)(23,25)(26,28)(27,31)(29,33)(30,32);(10,13)(11,15)(12,16)(14,17)(18,20)(19,23)(21,25)(22,24)(26,27)(28,30)(29,32)(31,33)

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 7337);