D₉₁₉₁ #32913635

order	18382
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₁₉₁
unambiguous semidirect product	C₇⋊D₁₃₁₃
unambiguous semidirect product	C₁₃⋊D₇₀₇
unambiguous semidirect product	C₁₀₁⋊D₉₁

Splittings

Fitting, socle, derived, Hall	C₇×C₁₃×C₁₀₁ ⋊ C₂
Hall	C₁₃×C₁₀₁ ⋊ D₇
Hall	C₇×C₁₀₁ ⋊ D₁₃
Sylow	C₁₀₁ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₁₀₁
Sylow	C₁₃ ⋊ D₇₀₇
Sylow	C₇ ⋊ D₁₃₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₁₀₁	C₂
LCS	G ▷ C₇×C₁₃×C₁₀₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 9191

A minimal faithful permutation representation — 2 generators on 121 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2,7,13,101|b^a=b^6,c^a=c^12,d^a=d^100

Conjugacy classes & character table

4597 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₁×D₁₀₁, D₁₃×D₇₀₇, D₇×D₁₃₁₃, …
maximal normal subgroup of	D₉₁×D₁₀₁, D₁₃×D₇₀₇, D₇×D₁₃₁₃, …
maximal quotient of	Dic₉₁₉₁, C₂×D₉₁₉₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18382, 8);

D9191 #32913635