C₅×#7703 #11930423

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7703

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

394 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 1280

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

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

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, 691);