C₅×#19056 #11941776

order	1280
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#19056

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3549	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

992 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 222 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 — 4 generators on 37 points:

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

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, 12044);