C₅×#18997 #11941717

order	1280
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#18997

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×H(3,𝔽₄)	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 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 — 4 generators on 37 points:

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

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, 11985);