D₉₂₆₉ #32915151

order	18538
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₂₆₉
unambiguous semidirect product	C₁₃⋊D₇₁₃
unambiguous semidirect product	C₂₃⋊D₄₀₃
unambiguous semidirect product	C₃₁⋊D₂₉₉

Splittings

Fitting, socle, derived, Hall	C₁₃×C₂₃×C₃₁ ⋊ C₂
Hall	C₂₃×C₃₁ ⋊ D₁₃
Hall	C₁₃×C₃₁ ⋊ D₂₃
Hall	C₁₃×C₂₃ ⋊ D₃₁
Sylow	C₃₁ ⋊ D₂₉₉
Sylow	C₂₃ ⋊ D₄₀₃
Sylow	C₁₃ ⋊ D₇₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₂₃×C₃₁	C₂
LCS	G ▷ C₁₃×C₂₃×C₃₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 9269

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

Polycyclic presentation:

2,13,23,31|b^a=b^12,c^a=c^22,d^a=d^30

Conjugacy classes & character table

4636 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁×D₂₉₉, D₂₃×D₄₀₃, D₁₃×D₇₁₃, …
maximal normal subgroup of	D₃₁×D₂₉₉, D₂₃×D₄₀₃, D₁₃×D₇₁₃, …
maximal quotient of	Dic₉₂₆₉, C₂×D₉₂₆₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18538, 8);

D9269 #32915151