C₅×#12230 #11934950

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#12230

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅
prime radical	C₅×#392	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₂²⋊C₄ ◁ G

422 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 160

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

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5218);