C₅×#85455 #14511193

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

solvable

Aliases

direct product

C₅×#85455

Splittings

Fitting

C₅×#568 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×C₄⋊S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#568	D₃
prime radical	C₅×#5923	C₂
hypercenter	C₅×M₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×M₁₆

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

normal subgroups only

This group has 486 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,56,47,23,11,55,8,52,34,9,20,66,13,24,49,19,32,58,10,44,31,26,28,67)(7,61,63,22,16,64,27,50,43,15,48,69,18,39,62,38,33,65,14,57,42,41,36,68)(12,25,29,53,35,59,21,45)(17,40,37,51,30,60,46,54);(6,8,13,10)(7,14,18,27)(9,23,26,19)(11,20,32,28)(12,21,35,29)(15,38,41,22)(16,36,33,48)(17,37,30,46)(24,52,56,44)(25,53,59,45)(31,47,34,49)(39,57,61,50)(40,54,60,51)(42,62,43,63)(55,67,58,66)(64,69,65,68);(6,7)(8,14)(9,22)(10,27)(11,30)(12,33)(13,18)(15,23)(16,35)(17,32)(19,41)(20,46)(21,48)(24,54)(25,57)(26,38)(28,37)(29,36)(31,43)(34,42)(39,45)(40,44)(47,63)(49,62)(50,59)(51,56)(52,60)(53,61)(55,69)(58,68)(64,66)(65,67)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236891);