D₈₀₉₉ #30633737

order	16198
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₀₉₉
unambiguous semidirect product	C₇⋊D₁₁₅₇
unambiguous semidirect product	C₁₃⋊D₆₂₃
unambiguous semidirect product	C₈₉⋊D₉₁

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₈₉ ⋊ C₂
Hall	C₁₃×C₈₉ ⋊ D₇
Hall	C₇×C₈₉ ⋊ D₁₃
Hall	C₇×C₁₃ ⋊ D₈₉
Sylow	C₈₉ ⋊ D₉₁
Sylow	C₁₃ ⋊ D₆₂₃
Sylow	C₇ ⋊ D₁₁₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₈₉	C₂
LCS	G ▷ C₇×C₁₃×C₈₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 8099

A minimal faithful permutation representation — 2 generators on 109 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,109)(23,108)(24,107)(25,106)(26,105)(27,104)(28,103)(29,102)(30,101)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)

Polycyclic presentation:

2,7,13,89|b^a=b^6,c^a=c^12,d^a=d^88

Conjugacy classes & character table

4051 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₈₉×D₉₁, D₁₃×D₆₂₃, D₇×D₁₁₅₇, …
maximal normal subgroup of	D₈₉×D₉₁, D₁₃×D₆₂₃, D₇×D₁₁₅₇, …
maximal quotient of	Dic₈₀₉₉, C₂×D₈₀₉₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16198, 8);

D8099 #30633737