C₅×#8788 #11931508

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#8788

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

322 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 158 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)(8,11)(9,10)(12,13)(14,15)(16,24)(17,20)(18,28)(19,23)(21,26)(22,25)(27,29)(30,31,33,36,34,37,40,43)(32,35,39,42,38,41,44,45);(7,10)(11,13)(14,16,25,27,19,26,18,17)(15,20,28,21,23,29,22,24)(30,32)(31,35)(33,39)(34,38)(36,42)(37,41)(40,44)(43,45);(6,8,9,12)(7,11,10,13)(14,17,18,26,19,27,25,16)(15,21,22,20,23,24,28,29)(32,38)(35,41)(39,44)(42,45)

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1776);