D₁₆₆₇₇ #45688666

order	33354
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₆₇₇
unambiguous semidirect product	C₉⋊D₁₈₅₃
unambiguous semidirect product	C₁₇⋊D₉₈₁
unambiguous semidirect product	C₁₀₉⋊D₁₅₃

Splittings

Fitting, derived, Hall	C₉×C₁₇×C₁₀₉ ⋊ C₂
Hall	C₁₇×C₁₀₉ ⋊ D₉
Hall	C₉×C₁₀₉ ⋊ D₁₇
Hall	C₉×C₁₇ ⋊ D₁₀₉
Sylow	C₁₀₉ ⋊ D₁₅₃
Sylow	C₁₇ ⋊ D₉₈₁
Sylow	C₉ ⋊ D₁₈₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₁₇×C₁₀₉	C₂
Frattini	C₃	D₅₅₅₉
socle	C₃×C₁₇×C₁₀₉	D₃
LCS	G ▷ C₉×C₁₇×C₁₀₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 16677

A minimal faithful permutation representation — 2 generators on 135 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(2,9)(3,8)(4,7)(5,6)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2,3,17,109,3|b^3=e^2,b^a=b^2e,c^a=c^16,d^a=d^108,e^a=e^2

Conjugacy classes & character table

8340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33354, 9);

D16677 #45688666