C₅×#10225 #11932945

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#10225

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

574 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 320

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

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

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, 3213);