C₅×#67536 #14278008

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#67536

Splittings

Hall	C₃×C₅ ⋊ #1340
Sylow	C₃ ⋊ C₅×#1340

Subgroups

	subgroup	G/H
center	C₂×C₅	#4470
commutator	C₂×C₃×C₈	C₂×C₄×C₅
Frattini	C₂×M₁₆	C₂×C₅×D₃
Fitting	C₃×C₅×#349	C₂
socle	C₂×C₃×C₅	#327
prime radical	C₂³×C₃×C₅	D₈
cosocle	C₂×C₃×M₁₆	C₂²×C₅
hypercenter	C₅×#349	D₃
hypocenter	C₃	C₅×#1340
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×M₁₆ ◁ C₅×#349

208 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 40
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,4,5)(6,7)(9,11,15,22)(10,16,19,32)(12,26,27,38)(13,24,14,23)(17,33,34,40)(18,35,30,31)(20,36,37,25)(21,28,39,29);(7,8)(9,10)(11,20,22,37)(12,25,27,36)(13,18)(14,30)(15,19)(16,21)(17,28,34,29)(23,33,24,40)(26,35)(31,38)(32,39)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 3706);