C₅×#85466 #14511204

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

solvable

Aliases

direct product

C₅×#85466

Splittings

Fitting

C₅×#576 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	D₄×S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#576	D₃
hypercenter	C₅×Q₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×Q₁₆

490 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 64 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)(6,48,29,22,30,49,12,20,31,8,10,47)(7,60,39,34,40,61,17,32,41,13,15,59)(9,44,55,19,56,68,27,52,57,24,25,69)(11,21,28,46)(14,42,64,18,50,66,23,62,65,35,36,67)(16,33,38,58)(26,53,54,45)(37,63,51,43);(6,7,12,17)(8,18,22,35)(9,23,27,14)(10,15,30,40)(11,16,28,38)(13,24,34,19)(20,42,48,62)(21,43,46,63)(25,50,56,36)(26,51,54,37)(29,39,31,41)(32,52,60,44)(33,53,58,45)(47,66,49,67)(55,65,57,64)(59,69,61,68);(10,28)(11,30)(15,38)(16,40)(20,46)(21,48)(25,54)(26,56)(29,31)(32,58)(33,60)(36,51)(37,50)(39,41)(42,63)(43,62)(44,53)(45,52)(47,49)(55,57)(59,61)(64,65)(66,67)(68,69)

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, 236902);