D₁₉⋊C₃×D₅₉ #27212103

order	13452
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₁₉⋊C₃×D₅₉
unambiguous semidirect product	D₁₉⋊(C₃×D₅₉)
unambiguous semidirect product	(D₁₉×D₅₉)⋊C₃
unambiguous semidirect product	(C₁₉⋊C₃×D₅₉)⋊C₂
unambiguous semidirect product	(C₅₉×D₁₉⋊C₃)⋊C₂
unambiguous semidirect product	((C₁₉⋊C₃)⋊D₅₉)⋊C₂

Splittings

Fitting, socle, derived, Hall	C₁₉×C₅₉ ⋊ C₂²×C₃
Hall	D₁₉×D₅₉ ⋊ C₃
Hall	C₁₉⋊C₃×C₅₉ ⋊ C₂²
Sylow	C₅₉ ⋊ C₂×D₁₉⋊C₃
Hall	C₁₉⋊C₃ ⋊ C₂×D₅₉
Sylow	C₁₉ ⋊ C₂×C₃×D₅₉

Subgroups

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

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 1121

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13452, 14);

D19⋊C3×D59 #27212103