D₃×D₁₉×C₅₉ #27212110

order	13452
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×D₁₉×C₅₉
unambiguous semidirect product	(D₃×C₁₉)⋊(C₂×C₅₉)
unambiguous semidirect product	(C₃×D₁₉)⋊(C₂×C₅₉)
unambiguous semidirect product	D₅₇⋊(C₂×C₅₉)

Splittings

Fitting, socle, Hall	C₃×C₁₉×C₅₉ ⋊ C₂²
derived, Hall	C₃×C₁₉ ⋊ C₂²×C₅₉
Hall	C₁₉×C₅₉ ⋊ C₂×D₃
Hall	C₃×C₅₉ ⋊ C₂×D₁₉
Sylow	C₁₉ ⋊ C₂×D₃×C₅₉
Sylow	C₃ ⋊ C₂×D₁₉×C₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅₉	D₃×D₁₉
commutator, cosocle, hypocenter	C₃×C₁₉	C₂²×C₅₉
Frattini	C₁	G
Fitting, socle	C₃×C₁₉×C₅₉	C₂²
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₅₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3363

A minimal faithful permutation representation — 2 generators on 81 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(21,22)

Conjugacy classes & character table

1947 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13452, 21);

D3×D19×C59 #27212110