C₅×#8188 #11930908

order	1280
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#8188

Subgroups

	subgroup	G/H
center	C₄²×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

412 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 640

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 1176);