#91270

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

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅ ⋊ #135
Sylow	C₅ ⋊ #904
Sylow	C₃ ⋊ #4058

Subgroups

	subgroup	G/H
center	C₂	D₅×D₁₂
commutator, cosocle	C₃×C₄×C₅	C₂³
Frattini	C₄	C₂×D₃×D₅
Fitting	C₃×C₅×C₈	C₂²
socle	C₂×C₃×C₅	C₂×D₄
hypercenter	C₈	D₃×D₅
hypocenter	C₃×C₅	#135
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

124 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^5,3,5|c^2=de,d^2=e,b^a=be,g^a=g^4,c^b=cd,d^b=de,f^b=f^2

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746460, #11746462, #11746464, …
maximal normal subgroup of	#11746460, #11746462, #11746464, …
maximal quotient of	#11747672, #11747673, #11747744, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 346);