D₅₄₉₉ #24926951

order	10998
automorphism	Aut(G) order 18212688
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₄₉₉
unambiguous semidirect product	C₉⋊D₆₁₁
unambiguous semidirect product	C₁₃⋊D₄₂₃
unambiguous semidirect product	C₄₇⋊D₁₁₇

Splittings

Fitting, derived, Hall	C₉×C₁₃×C₄₇ ⋊ C₂
Hall	C₁₃×C₄₇ ⋊ D₉
Hall	C₉×C₄₇ ⋊ D₁₃
Hall	C₉×C₁₃ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₁₁₇
Sylow	C₁₃ ⋊ D₄₂₃
Sylow	C₉ ⋊ D₆₁₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₃×C₄₇	C₂
Frattini	C₃	D₁₈₃₃
socle	C₃×C₁₃×C₄₇	D₃
LCS	G ▷ C₉×C₁₃×C₄₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 5499

A minimal faithful permutation representation — 2 generators on 69 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);(2,9)(3,8)(4,7)(5,6)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2,3,13,47,3|b^3=e^2,b^a=b^2e,c^a=c^12,d^a=d^46,e^a=e^2

Conjugacy classes & character table

2751 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₇×D₁₁₇, D₁₃×D₄₂₃, D₉×D₆₁₁, …
maximal normal subgroup of	D₄₇×D₁₁₇, D₁₃×D₄₂₃, D₉×D₆₁₁, …
maximal quotient of	Dic₅₄₉₉, C₂×D₅₄₉₉, D₁₆₄₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10998, 9);

D5499 #24926951