D₁₃₄₀₉ #41001210

order	26818
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₄₀₉
unambiguous semidirect product	C₁₁⋊D₁₂₁₉
unambiguous semidirect product	C₂₃⋊D₅₈₃
unambiguous semidirect product	C₅₃⋊D₂₅₃

Splittings

Fitting, socle, derived, Hall	C₁₁×C₂₃×C₅₃ ⋊ C₂
Hall	C₂₃×C₅₃ ⋊ D₁₁
Hall	C₁₁×C₅₃ ⋊ D₂₃
Hall	C₁₁×C₂₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₂₅₃
Sylow	C₂₃ ⋊ D₅₈₃
Sylow	C₁₁ ⋊ D₁₂₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₂₃×C₅₃	C₂
LCS	G ▷ C₁₁×C₂₃×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 13409

A minimal faithful permutation representation — 2 generators on 87 points:

(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(2,11)(3,10)(4,9)(5,8)(6,7)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2,11,23,53|b^a=b^10,c^a=c^22,d^a=d^52

Conjugacy classes & character table

6706 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₄₀₉
maximal normal subgroup of	C₂×D₁₃₄₀₉
maximal quotient of	Dic₁₃₄₀₉, C₂×D₁₃₄₀₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26818, 9);

D13409 #41001210