C₅×D₁₃₀₅ #27189404

order	13050
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₃₀₅
unambiguous semidirect product	C₅⋊(C₅×D₂₆₁)
unambiguous semidirect product	C₉⋊(C₅×D₁₄₅)
unambiguous semidirect product	C₂₉⋊(C₅×D₄₅)

Splittings

Fitting, Hall	C₅²×C₉×C₂₉ ⋊ C₂
derived	C₅×C₉×C₂₉ ⋊ C₂×C₅
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	C₅	D₁₃₀₅
commutator, cosocle, hypocenter	C₅×C₉×C₂₉	C₂×C₅
Frattini	C₃	C₅×D₄₃₅
Fitting	C₅²×C₉×C₂₉	C₂
socle	C₃×C₅²×C₂₉	D₃
LCS	G ▷ C₅×C₉×C₂₉
UCS	C₁ ◁ C₅

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 2610

A minimal faithful permutation representation — 2 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)(30,31,32,33,34)(36,37)(38,39)(41,43)(42,46)(44,45)(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)(35,36,38,39,37)(40,48,43,46,45,44,42,41,47)

Conjugacy classes & character table

3270 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₂₉×D₄₅, C₅×D₅×D₂₆₁, C₅×D₉×D₁₄₅, …
maximal normal subgroup of	C₅×D₂₉×D₄₅, C₅×D₅×D₂₆₁, C₅×D₉×D₁₄₅, …
maximal quotient of	C₅×Dic₁₃₀₅, C₂×C₅×D₁₃₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13050, 36);

C5×D1305 #27189404