D₃×C₁₃⋊C₃×C₅₃ #27137166

order	12402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×C₁₃⋊C₃×C₅₃
unambiguous semidirect product	C₃⋊(C₂×C₁₃⋊C₃×C₅₃)
unambiguous semidirect product	(D₃×C₁₃)⋊(C₃×C₅₃)
unambiguous semidirect product	(D₃×C₁₃×C₅₃)⋊C₃

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅₃	D₃×C₁₃⋊C₃
commutator, cosocle, hypocenter	C₃×C₁₃	C₂×C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃×C₅₃	C₂×C₃
LCS	G ▷ C₃×C₁₃
UCS	C₁ ◁ C₅₃

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:

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

Conjugacy classes & character table

1113 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₅₃×D₁₃⋊C₃, D₃×(C₁₃⋊C₃)⋊D₅₃, D₃×C₁₃⋊C₃×D₅₃, …
maximal normal subgroup of	D₃×C₅₃×D₁₃⋊C₃, D₃×(C₁₃⋊C₃)⋊D₅₃, D₃×C₁₃⋊C₃×D₅₃, …
maximal quotient of	Dic₃×C₁₃⋊C₃×C₅₃, C₂×D₃×C₁₃⋊C₃×C₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12402, 29);

D3×C13⋊C3×C53 #27137166