C₅×#12724 #11935444

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12724

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1782
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₄×C₅×#41 ◁ G

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

normal subgroups only

This group has 670 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)(8,17)(9,21,23,14)(10,15)(11,16)(12,18,28,35)(13,22,29,37)(19,32)(20,33)(24,26)(25,30)(27,34,31,36);(6,8,11,20)(7,12,16,28)(9,18,23,35)(10,24,25,19)(13,26,29,32)(14,17,21,33)(15,31,30,27)(22,36,37,34);(6,9,11,23)(7,13)(8,18,20,35)(10,22,25,37)(12,26)(14,30)(15,21)(16,29)(17,27)(19,34,24,36)(28,32)(31,33)

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, 5712);