C₅×#7650 #11930370

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7650

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

384 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 45
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7,9,11,10,12,8,13)(14,15)(16,22,24,19)(17,25)(18,21)(20,26)(23,27,29,28)(30,31)(32,38,40,45)(33,37)(34,36)(35,39,43,41)(42,44);(14,16,18,24)(15,19,21,22)(17,23,26,29)(20,27,25,28)(30,32)(31,35)(33,41)(34,40)(36,43)(37,45)(38,44)(39,42);(6,8,10,9)(7,11,12,13)(14,17,18,26)(15,20,21,25)(16,23,24,29)(19,27,22,28)(31,36)(35,43)(37,44)(38,45)

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, 638);