C₅⋊D₁₃₀₅ #27189406

order	13050
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₅

Aliases

unambiguous semidirect product	C₅⋊D₁₃₀₅
unambiguous semidirect product	C₉⋊(C₅⋊D₁₄₅)
unambiguous semidirect product	C₂₉⋊(C₅⋊D₄₅)

Splittings

Fitting, derived, Hall	C₅²×C₉×C₂₉ ⋊ C₂
Hall	C₅²×C₂₉ ⋊ D₉
Hall	C₉×C₂₉ ⋊ C₅⋊D₅
Hall	C₅²×C₉ ⋊ D₂₉
Sylow	C₂₉ ⋊ C₅⋊D₄₅
Sylow	C₅² ⋊ D₂₆₁
Sylow	C₉ ⋊ C₅⋊D₁₄₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅²×C₉×C₂₉	C₂
Frattini	C₃	C₃⋊(C₅⋊D₁₄₅)
socle	C₃×C₅²×C₂₉	D₃
LCS	G ▷ C₅²×C₉×C₂₉
UCS	C₁

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 6525

A minimal faithful permutation representation — 3 generators on 48 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(31,33)(32,36)(34,35)(37,38)(40,41)(42,43)(45,46)(47,48);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(30,38,33,36,35,34,32,31,37)(44,45,47,48,46);(39,40,42,43,41)

Polycyclic presentation:

2,3,5^2,29,3|b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^4,e^a=e^28,f^a=f^2

Conjugacy classes & character table

3264 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉×C₅⋊D₄₅, C₅⋊D₅×D₂₆₁, D₉×C₅⋊D₁₄₅, …
maximal normal subgroup of	D₂₉×C₅⋊D₄₅, C₅⋊D₅×D₂₆₁, D₉×C₅⋊D₁₄₅, …
maximal quotient of	C₉⋊#15668515, C₂×C₅⋊D₁₃₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 38);

C5⋊D1305 #27189406