D₉₀₆₃ #31827369

order	18126
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₀₆₃
unambiguous semidirect product	C₉⋊D₁₀₀₇
unambiguous semidirect product	C₁₉⋊D₄₇₇
unambiguous semidirect product	C₅₃⋊D₁₇₁

Splittings

Fitting, derived, Hall	C₉×C₁₉×C₅₃ ⋊ C₂
Hall	C₁₉×C₅₃ ⋊ D₉
Hall	C₉×C₅₃ ⋊ D₁₉
Hall	C₉×C₁₉ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₁₇₁
Sylow	C₁₉ ⋊ D₄₇₇
Sylow	C₉ ⋊ D₁₀₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₉×C₅₃	C₂
Frattini	C₃	D₃₀₂₁
socle	C₃×C₁₉×C₅₃	D₃
LCS	G ▷ C₉×C₁₉×C₅₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 9063

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

Polycyclic presentation:

2,3,19,53,3|b^3=e^2,b^a=b^2e,c^a=c^18,d^a=d^52,e^a=e^2

Conjugacy classes & character table

4533 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₃×D₁₇₁, D₁₉×D₄₇₇, D₉×D₁₀₀₇, …
maximal normal subgroup of	D₅₃×D₁₇₁, D₁₉×D₄₇₇, D₉×D₁₀₀₇, …
maximal quotient of	Dic₉₀₆₃, C₂×D₉₀₆₃, D₂₇₁₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18126, 9);

D9063 #31827369