D₁₇₈₈₉ #49088765

order	35778
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₈₈₉
unambiguous semidirect product	C₃⋊D₅₉₆₃
unambiguous semidirect product	C₆₇⋊D₂₆₇
unambiguous semidirect product	C₈₉⋊D₂₀₁

Splittings

Fitting, socle, derived, Hall	C₃×C₆₇×C₈₉ ⋊ C₂
Hall	C₆₇×C₈₉ ⋊ D₃
Hall	C₃×C₈₉ ⋊ D₆₇
Hall	C₃×C₆₇ ⋊ D₈₉
Sylow	C₈₉ ⋊ D₂₀₁
Sylow	C₆₇ ⋊ D₂₆₇
Sylow	C₃ ⋊ D₅₉₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₆₇×C₈₉	C₂
LCS	G ▷ C₃×C₆₇×C₈₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 159
transitive degree	μ_t(G) = 17889

A minimal faithful permutation representation — 2 generators on 159 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)(71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(2,3)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(72,159)(73,158)(74,157)(75,156)(76,155)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143)(89,142)(90,141)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)

Polycyclic presentation:

2,3,67,89|b^a=b^2,c^a=c^66,d^a=d^88

Conjugacy classes & character table

8946 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35778, 9);

D17889 #49088765