C₅×#19006 #11941726

order	1280
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#19006

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×#562	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

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

normal subgroups only

This group has 938 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)(9,15)(10,14)(11,13)(12,17)(18,21)(19,20)(22,23,26,28)(24,30,32,37)(25,33,35,29)(27,34,36,31);(6,8)(7,12)(9,19)(10,17)(11,18)(13,20)(14,21)(15,16)(22,24,26,32)(23,27,28,36)(25,34,35,31)(29,37,33,30);(6,8)(7,12)(9,17)(10,19)(11,18)(13,20)(14,16)(15,21)(25,35)(29,33)(30,37)(31,34);(7,13)(12,20)(14,15)(16,21)(23,28)(27,36)(29,33)(30,37)

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, 11994);