D₁₁₃₄₉ #36384111

order	22698
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₃₄₉
unambiguous semidirect product	C₉⋊D₁₂₆₁
unambiguous semidirect product	C₁₃⋊D₈₇₃
unambiguous semidirect product	C₉₇⋊D₁₁₇

Splittings

Fitting, derived, Hall	C₉×C₁₃×C₉₇ ⋊ C₂
Hall	C₁₃×C₉₇ ⋊ D₉
Hall	C₉×C₉₇ ⋊ D₁₃
Hall	C₉×C₁₃ ⋊ D₉₇
Sylow	C₉₇ ⋊ D₁₁₇
Sylow	C₁₃ ⋊ D₈₇₃
Sylow	C₉ ⋊ D₁₂₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₃×C₉₇	C₂
Frattini	C₃	D₃₇₈₃
socle	C₃×C₁₃×C₉₇	D₃
LCS	G ▷ C₉×C₁₃×C₉₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 11349

A minimal faithful permutation representation — 2 generators on 119 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);(2,9)(3,8)(4,7)(5,6)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(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:

2,3,13,97,3|b^3=e^2,b^a=b^2e,c^a=c^12,d^a=d^96,e^a=e^2

Conjugacy classes & character table

5676 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22698, 12);

D11349 #36384111