C₅×#85360 #14511098

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85360

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#520	D₃
prime radical	C₂×C₅×#41⋊C₃	C₂²
cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
hypercenter	C₅×C₄⋊C₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

510 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 41
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,8,9,7)(10,23,28)(11,27,33)(12,21,39,20,34,37)(13,14,26)(16,30,41,29,38,40)(17,18,32)(22,35)(31,36);(10,12,13,20)(11,16,17,29)(14,21,23,34)(15,22,24,35)(18,30,27,38)(19,31,25,36)(26,37,28,39)(32,40,33,41);(6,7,9,8)(10,11,13,17)(12,16,20,29)(14,25,23,19)(15,27,24,18)(21,36,34,31)(22,38,35,30)(26,33,28,32)(37,41,39,40)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236796);