C₃×C₁₉×D₂₉ #15693249

order	3306
isoclinism	isoclinic class of D₂₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₁₉×D₂₉
unambiguous semidirect product	C₂₉⋊(C₂×C₃×C₁₉)

Splittings

Fitting, socle, Hall	C₃×C₁₉×C₂₉ ⋊ C₂
derived, Sylow	C₂₉ ⋊ C₂×C₃×C₁₉
Hall	C₁₉×C₂₉ ⋊ C₂×C₃
Hall	C₃×C₂₉ ⋊ C₂×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₁₉	D₂₉
commutator, cosocle, hypocenter	C₂₉	C₂×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₂₉	C₂
LCS	G ▷ C₂₉
UCS	C₁ ◁ C₃×C₁₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1653

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₅×C₁₉×D₂₉, C₃×C₁₉×D₁₄₅, C₃×C₇×C₁₉×D₂₉, …
maximal normal subgroup of	C₃×C₅×C₁₉×D₂₉, C₃×C₇×C₁₉×D₂₉, C₃×C₁₉×D₂₉⋊C₇, …
maximal quotient of	C₃×C₅×C₁₉×D₂₉, C₃×C₁₉×D₁₄₅, C₃×C₇×C₁₉×D₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3306, 3);

C3×C19×D29 #15693249