C₅×#85542 #14511280

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

solvable

Aliases

direct product

C₅×#85542

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×GL(2,ℤ/4)
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#585	D₃
prime radical	C₅×#5945	C₂
hypercenter	C₅×#41	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

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

normal subgroups only

This group has 572 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)(6,28,30)(7,39,42,14,29,63)(8,21,51,19,44,49)(9,24,55,23,52,54)(10,27,12)(13,60,59,32,50,69)(15,16,62,18,37,41)(17,38)(20,64,67)(22,45)(25,53)(26,40)(33,68,35,56,58,36)(34,57)(43,46,66)(48,61);(6,8)(7,13)(9,20)(10,21)(11,22)(12,19)(14,36)(15,32)(16,33)(17,34)(18,35)(23,43)(24,46)(25,47)(26,48)(27,49)(28,44)(29,50)(30,51)(31,45)(37,60)(38,61)(39,56)(40,57)(41,58)(42,59)(52,64)(53,65)(54,66)(55,67)(62,69)(63,68);(6,7,12,18)(8,13,19,35)(9,14,23,15)(10,26,28,17)(11,29,31,16)(20,36,43,32)(21,48,44,34)(22,50,45,33)(24,40,52,38)(25,39,53,37)(27,42,30,41)(46,57,64,61)(47,56,65,60)(49,59,51,58)(54,63,55,62)(66,68,67,69)

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, 236978);