D₁₆₆₈₅ #45688760

order	33370
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₆₈₅
unambiguous semidirect product	C₅⋊D₃₃₃₇
unambiguous semidirect product	C₄₇⋊D₃₅₅
unambiguous semidirect product	C₇₁⋊D₂₃₅

Splittings

Fitting, socle, derived, Hall	C₅×C₄₇×C₇₁ ⋊ C₂
Hall	C₄₇×C₇₁ ⋊ D₅
Hall	C₅×C₇₁ ⋊ D₄₇
Hall	C₅×C₄₇ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₂₃₅
Sylow	C₄₇ ⋊ D₃₅₅
Sylow	C₅ ⋊ D₃₃₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₄₇×C₇₁	C₂
LCS	G ▷ C₅×C₄₇×C₇₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 16685

A minimal faithful permutation representation — 2 generators on 123 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);(2,5)(3,4)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)

Polycyclic presentation:

2,5,47,71|b^a=b^4,c^a=c^46,d^a=d^70

Conjugacy classes & character table

8344 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33370, 9);

D16685 #45688760