C₅×#909 #91712

order	480
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#909

Splittings

Fitting	C₃×C₅×M₁₆ ⋊ C₂
Hall	C₃×C₅ ⋊ #137
Sylow	C₃ ⋊ C₅×#137

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×D₁₂
commutator, cosocle	C₃×C₄	C₂³×C₅
Frattini	C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×M₁₆	C₂
socle	C₂×C₃×C₅	C₂×D₄
prime radical	C₅×#235	C₂
hypercenter	C₅×M₁₆	D₃
hypocenter	C₃	C₅×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×M₁₆

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3)(4,16,8,13,9,7,6,15)(5,19,10,18,12,11,17,14)(20,21,22,23,24);(4,6,9,8)(5,10,12,17)(7,15,16,13)(11,18,19,14);(2,3)(4,5)(6,10)(7,14)(8,17)(9,12)(11,15)(13,19)(16,18)

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11748186, #11748188, #11748212, …
maximal normal subgroup of	#11748186, #11748188, #11748212, …
maximal quotient of	C₅×#4702, C₅×#4703, C₅×#4710, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 788);