C₅₉×(C₁₉⋊C₃)⋊C₄ #27212100

order	13452
isoclinism	isoclinic class of D₁₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅₉×(C₁₉⋊C₃)⋊C₄
unambiguous semidirect product	(C₁₉⋊C₃)⋊(C₄×C₅₉)
unambiguous semidirect product	Dic₁₉⋊(C₃×C₅₉)
unambiguous semidirect product	(C₅₉×Dic₁₉)⋊C₃

Splittings

derived, Sylow	C₁₉ ⋊ C₃×C₄×C₅₉
Hall	C₅₉×Dic₁₉ ⋊ C₃
Hall	C₁₉⋊C₃×C₅₉ ⋊ C₄
Hall	C₁₉×C₅₉ ⋊ C₃×C₄
Hall	Dic₁₉ ⋊ C₃×C₅₉
Hall	C₁₉⋊C₃ ⋊ C₄×C₅₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅₉	D₁₉⋊C₃
commutator, hypocenter	C₁₉	C₃×C₄×C₅₉
Frattini	C₂	C₅₉×D₁₉⋊C₃
Fitting, socle	C₂×C₁₉×C₅₉	C₂×C₃
prime radical	C₂×C₁₉⋊C₃×C₅₉	C₂
cosocle	C₂×C₁₉	C₂×C₃×C₅₉
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₅₉

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 82
transitive degree	μ_t(G) = 4484

A minimal faithful permutation representation — 2 generators on 82 points:

(2,6,7,3,4,8)(5,14,11,9,10,16)(12,17,19,18,15,13)(20,23,22,21)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82);(1,2,5,11,17,14,18,4,8,13,19,7,6,12,10,15,16,9,3)

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, 11);

C59×(C19⋊C3)⋊C4 #27212100