C₅×#69379 #14319206

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#69379

Splittings

Fitting	C₃×C₅×SD₆₄ ⋊ C₂
Hall	C₃×C₅ ⋊ #2262
Sylow	C₃ ⋊ C₅×#2262

Subgroups

	subgroup	G/H
center	C₂×C₅	D₃×D₁₆
commutator, cosocle	C₃×C₁₆	C₂³×C₅
Frattini	C₁₆	C₂²×C₅×D₃
Fitting	C₃×C₅×SD₆₄	C₂
socle	C₂×C₃×C₅	C₂×D₁₆
hypercenter	C₅×SD₆₄	D₃
hypocenter	C₃	C₅×#2262
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×C₁₆ ◁ C₅×SD₆₄

212 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 72
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,3,4,5)(6,7)(9,12,15,30)(10,17,21,20)(11,23,27,26)(13,28,34,61)(14,29,37,62)(16,38,40,48)(18,41,45,44)(19,42,47,35)(22,49,50,57)(24,51,54,53)(25,52,56,36)(31,58,64,63)(32,59,66,65)(33,60,67,70)(39,46,68,72)(43,55,69,71);(9,11)(10,16)(12,23)(13,32)(14,36)(15,27)(17,38)(18,43)(19,46)(20,48)(21,40)(22,31)(24,33)(25,29)(26,30)(28,59)(34,66)(35,39)(37,52)(41,55)(42,68)(44,71)(45,69)(47,72)(49,58)(50,64)(51,60)(53,70)(54,67)(56,62)(57,63)(61,65);(7,8)(9,10)(11,22)(12,17)(13,31)(14,35)(15,21)(16,24)(18,33)(19,29)(20,30)(23,49)(25,55)(26,57)(27,50)(28,58)(32,46)(34,64)(36,43)(37,42)(38,51)(39,65)(40,54)(41,60)(44,70)(45,67)(47,62)(48,53)(52,69)(56,71)(59,68)(61,63)(66,72)

Conjugacy classes & character table

285 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 44904);