C₅×#7395 #11930115

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7394
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7395

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1343
commutator	C₂×C₂²⋊C₄	C₂×C₄×C₅
Frattini, cosocle	#381	C₂²×C₅
prime radical	C₅×#534	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂⁴×C₅ ◁ C₅×#381 ◁ G

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

normal subgroups only

This group has 284 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)(6,7,10,14,11,15,23,31)(8,16,18,32,19,28,35,37)(9,20,21,36,22,13,27,29)(12,24,25,17,26,33,30,34);(6,8)(7,12)(9,17)(10,18)(11,19)(13,28)(14,30)(15,26)(16,20)(21,24)(22,34)(23,35)(25,31)(27,33)(29,37)(32,36)

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, 383);