C₅×#14663 #11937383

order	1280
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₅ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₅×#14663

Subgroups

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

1462 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1462 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) = 37
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 4 generators on 37 points:

(1,2,3,4,5)(14,15,17,19)(16,18,20,21)(23,27)(28,35)(29,32)(36,37);(7,9)(10,11,12,13)(16,20)(18,21)(22,23,26,27)(24,28,31,35)(25,32,34,29)(30,37,33,36);(6,7)(8,9)(10,11)(12,13)(14,15)(16,18)(17,19)(20,21)(22,24,26,31)(23,28,27,35)(25,33,34,30)(29,37,32,36);(14,16)(15,18)(17,20)(19,21)(22,25,26,34)(23,29,27,32)(24,30,31,33)(28,36,35,37)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 7651);