C₅×#7427 #11930147

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7426
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7427

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1351
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₅
Frattini, cosocle	#381	C₂²×C₅
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂⁴×C₅ ◁ C₅×#381 ◁ G

214 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 415);