C₅×#85451 #14511189

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

solvable

Aliases

direct product

C₅×#85451

Splittings

Fitting

C₂×C₅×#131 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	C₄⋊S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂×C₅×#131	D₃
socle	C₂×C₅	C₂×C₄⋊S₄
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

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

normal subgroups only

This group has 492 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)(6,56,46,23,11,55,8,52,34,9,20,66,13,24,48,19,32,58,10,43,31,26,28,67)(7,61,63,22,16,64,27,49,42,15,47,69,18,38,62,51,33,65,14,57,41,40,36,68)(12,25,29,53,35,59,21,44)(17,39,37,50,30,60,45,54);(6,8,13,10)(7,14,18,27)(9,19,26,23)(11,20,32,28)(12,21,35,29)(15,22,40,51)(16,36,33,47)(17,37,30,45)(24,43,56,52)(25,44,59,53)(31,46,34,48)(38,49,61,57)(39,50,60,54)(41,62,42,63)(55,66,58,67)(64,68,65,69);(6,7)(8,14)(9,22)(10,27)(11,30)(12,33)(13,18)(15,23)(16,35)(17,32)(19,40)(20,45)(21,47)(24,54)(25,57)(26,51)(28,37)(29,36)(31,42)(34,41)(38,44)(39,43)(46,63)(48,62)(49,59)(50,56)(52,60)(53,61)(55,69)(58,68)(64,66)(65,67)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236887);