#44477645

order	30420
automorphism	?
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived, Hall	C₅×C₁₃² ⋊ C₄×C₉
Hall	C₅×C₁₃×C₁₃⋊C₉ ⋊ C₄
Hall	#16810002 ⋊ C₉
Hall	C₁₃×C₁₃⋊C₉ ⋊ AGL(1,𝔽₅)
Sylow	C₁₃² ⋊ C₉×AGL(1,𝔽₅)
Sylow	C₅ ⋊ #20258774

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₃	#23778463
commutator, hypocenter	C₅×C₁₃²	C₄×C₉
Fitting, socle	C₃×C₅×C₁₃²	C₃×C₄
prime radical, cosocle	C₃×C₅⋊(C₁₃⋊D₁₃)	C₂×C₃
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₃

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 585

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

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

Polycyclic presentation:

2,3,2,5,13^2,3|a^2=c,b^3=g,d^a=d^2,e^a=e^5,f^a=f^5,e^b=e^3,d^c=d^4,e^c=e^12,f^c=f^12

Conjugacy classes & character table

375 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 234);