C₅×#68132 #14288050

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

Aliases

direct product

C₅×#68132

Splittings

Fitting	C₃×C₅×#324 ⋊ C₂
Hall	C₃×C₅ ⋊ #1603
Sylow	C₃ ⋊ C₅×#1603

Subgroups

	subgroup	G/H
center	C₂²×C₅	#882
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#324	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₅×#930	C₂²
hypercenter	C₅×#324	D₃
hypocenter	C₃	C₅×#1603
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#324

488 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,4,5)(6,7,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(7,8)(13,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 13748);