C₅×#85544 #14511282

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

solvable

Aliases

direct product

C₅×#85544

Splittings

Fitting

C₅×#585 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	GL(2,ℤ/4)
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#585	D₃
socle	C₂×C₅	C₂×GL(2,ℤ/4)
hypercenter	C₅×#41	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

594 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 53
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,39,47,8,22,50)(7,26,37,12,31,40)(9,36,43,21,27,32)(10,52,20,19,46,16)(11,35,44,14,45,51)(13,34,48,17,25,49)(15,24,30,29,38,23)(18,42,53,28,33,41);(6,7,13,18)(8,12,17,28)(9,14,20,30)(10,15,27,35)(11,16,23,21)(19,29,36,45)(22,31,25,33)(24,32,44,52)(26,34,42,39)(37,48,53,47)(38,43,51,46)(40,49,41,50);(8,16)(9,20)(10,24)(11,28)(12,23)(14,30)(15,32)(17,21)(19,37)(22,40)(25,41)(26,46)(27,44)(29,48)(31,49)(33,50)(34,38)(35,52)(36,53)(39,51)(42,43)(45,47)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236980);