C₅×#12779 #11935499

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12779

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1790
commutator	C₂×Q₈	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

732 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 160

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

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

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, 5767);