#91913

order	480
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₅ ⋊ #131
Sylow	C₅ ⋊ #907
Sylow	C₃ ⋊ #4131

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂×C₃×C₅	C₂²×C₄
Fitting	C₃×C₄×C₅	C₂×C₄
prime radical	#6772	C₂
cosocle	C₃×Dic₅	C₂³
hypercenter	C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#131
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11751685, #11751686, #11751687, …
maximal normal subgroup of	#11751685, #11751686, #11751687, …
maximal quotient of	#11751770, #11751782, #11751783, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 989);