C₅×#12221 #11934941

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12096
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12221

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊C₄ ◁ G

392 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,7,11,18)(8,20,23,37)(9,26,28,36)(10,30,21,16)(12,32,13,19)(14,29,35,22)(15,31,34,27)(17,24,33,25);(6,8,12,24)(7,14,32,26)(9,21,22,31)(10,29,27,28)(11,23,13,25)(15,33,30,20)(16,37,34,17)(18,35,19,36);(6,9)(7,15)(8,21)(10,23)(11,28)(12,22)(13,29)(14,33)(16,19)(17,35)(18,34)(20,26)(24,31)(25,27)(30,32)(36,37)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5209);