C₁₃⋊C₃×D₁₅₉ #27137167

order	12402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃⋊C₃×D₁₅₉
unambiguous semidirect product	C₃⋊(C₁₃⋊C₃×D₅₃)
unambiguous semidirect product	C₅₃⋊(D₃×C₁₃⋊C₃)
unambiguous semidirect product	(C₁₃×D₁₅₉)⋊C₃

Splittings

Fitting, socle, derived	C₃×C₁₃×C₅₃ ⋊ C₂×C₃
Hall	C₃×C₁₃⋊C₃×C₅₃ ⋊ C₂
Hall	C₁₃×C₅₃ ⋊ C₃×D₃
Hall	C₃×C₁₃⋊C₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₃×C₁₃⋊C₃
Sylow	C₁₃ ⋊ C₃×D₁₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₃×C₅₃	C₂×C₃
LCS	G ▷ C₃×C₁₃×C₅₃
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 2067

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₁₃⋊C₃×D₁₅₉, (C₃⋊(D₁₃⋊C₃))⋊D₅₃, D₃×C₁₃⋊C₃×D₅₃, …
maximal normal subgroup of	D₁₃⋊C₃×D₁₅₉, (C₃⋊(D₁₃⋊C₃))⋊D₅₃, D₃×C₁₃⋊C₃×D₅₃, …
maximal quotient of	C₁₃⋊C₃×Dic₁₅₉, C₂×C₁₃⋊C₃×D₁₅₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 30);

C13⋊C3×D159 #27137167