C₂×C₅₉×D₁₉⋊C₃ #27212104

order	13452
isoclinism	isoclinic class of D₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₅₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₅₉×D₁₉⋊C₃
unambiguous semidirect product	D₁₉⋊(C₂×C₃×C₅₉)
unambiguous semidirect product	(C₁₉⋊C₃)⋊(C₂²×C₅₉)
unambiguous semidirect product	(C₂×D₁₉)⋊(C₃×C₅₉)
unambiguous semidirect product	(C₂×C₁₉⋊C₃)⋊(C₂×C₅₉)
unambiguous semidirect product	(C₂×D₁₉×C₅₉)⋊C₃

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅₉	D₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₂²×C₃×C₅₉
Frattini	C₁	G
Fitting, socle	C₂×C₁₉×C₅₉	C₂×C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₅₉

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 2242

A minimal faithful permutation representation — 2 generators on 80 points:

(2,6,7,3,4,8)(5,14,11,9,10,16)(12,17,19,18,15,13)(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);(1,2,5,11,17,14,18,4,8,13,19,7,6,12,10,15,16,9,3)(79,80)

Conjugacy classes & character table

1062 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13452, 15);

C2×C59×D19⋊C3 #27212104