#91264

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

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅ ⋊ C₂×Q₁₆
Sylow	C₅ ⋊ D₃×Q₁₆
Sylow	C₃ ⋊ D₅×Q₁₆

Subgroups

	subgroup	G/H
center	C₂	#6777
commutator, cosocle	C₃×C₄×C₅	C₂³
Frattini	C₄	C₂×D₃×D₅
Fitting	C₃×C₅×C₈	C₂²
socle	C₂×C₃×C₅	C₂×D₄
prime radical	C₂×D₁₅	D₄
hypercenter	C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×Q₁₆
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

120 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)(9,12,21,24,14,23,13,22)(10,15,18,20,11,17,16,19);(7,8)(9,11,14,10)(12,20,23,19)(13,16,21,18)(15,22,17,24);(2,3)(4,5)(9,10,14,11)(12,19,23,20)(13,18,21,16)(15,24,17,22)

Polycyclic presentation:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11746823, #11746831, #11746841, …
maximal normal subgroup of	#11746823, #11746831, #11746841, …
maximal quotient of	#11747609, #11747636, #11747639, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 340);