C₅×#23470 #11946190

order	1280
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#23470

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

1178 conjugacy classes of subgroups; only the 334 normal subgroups are shown.

normal subgroups only

This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 334 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 — 4 generators on 45 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 16458);