C₅×#12709 #11935429

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12709

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1781
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂²×Q₈	C₂³×C₅
prime radical	C₅×#3485	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₅×#41 ◁ G

986 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 146 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)(6,7,10,14,11,15,26,33)(8,16,19,34,18,30,24,36)(9,20,23,32,22,13,25,31)(12,27,29,21,28,35,37,17);(6,8)(7,12)(9,21)(10,24)(11,18)(13,30)(14,29)(15,28)(16,20)(17,22)(19,26)(23,27)(25,35)(31,36)(32,34)(33,37);(9,22)(13,20)(16,30)(17,21)(23,25)(27,35)(31,32)(34,36)

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, 5697);