C₅×#8126 #11930846

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#8126

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#95
commutator	C₂³	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂⁴×C₅	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄²×C₅ ◁ G

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 1114);