D₁₁₁₆₉ #36375235

order	22338
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₁₆₉
unambiguous semidirect product	C₉⋊D₁₂₄₁
unambiguous semidirect product	C₁₇⋊D₆₅₇
unambiguous semidirect product	C₇₃⋊D₁₅₃

Splittings

Fitting, derived, Hall	C₉×C₁₇×C₇₃ ⋊ C₂
Hall	C₁₇×C₇₃ ⋊ D₉
Hall	C₉×C₇₃ ⋊ D₁₇
Hall	C₉×C₁₇ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₅₃
Sylow	C₁₇ ⋊ D₆₅₇
Sylow	C₉ ⋊ D₁₂₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₇×C₇₃	C₂
Frattini	C₃	D₃₇₂₃
socle	C₃×C₁₇×C₇₃	D₃
LCS	G ▷ C₉×C₁₇×C₇₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 11169

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

Polycyclic presentation:

2,3,17,73,3|b^3=e^2,b^a=b^2e,c^a=c^16,d^a=d^72,e^a=e^2

Conjugacy classes & character table

5586 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22338, 9);

D11169 #36375235