D₁₁₆₀₉ #37551059

order	23218
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₆₀₉
unambiguous semidirect product	C₁₃⋊D₈₉₃
unambiguous semidirect product	C₁₉⋊D₆₁₁
unambiguous semidirect product	C₄₇⋊D₂₄₇

Splittings

Fitting, socle, derived, Hall	C₁₃×C₁₉×C₄₇ ⋊ C₂
Hall	C₁₉×C₄₇ ⋊ D₁₃
Hall	C₁₃×C₄₇ ⋊ D₁₉
Hall	C₁₃×C₁₉ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₂₄₇
Sylow	C₁₉ ⋊ D₆₁₁
Sylow	C₁₃ ⋊ D₈₉₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₁₉×C₄₇	C₂
LCS	G ▷ C₁₃×C₁₉×C₄₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 11609

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

Polycyclic presentation:

2,13,19,47|b^a=b^12,c^a=c^18,d^a=d^46

Conjugacy classes & character table

5806 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23218, 8);

D11609 #37551059