D₉₇₁₇ #34085487

order	19434
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₇₁₇
unambiguous semidirect product	C₃⋊D₃₂₃₉
unambiguous semidirect product	C₄₁⋊D₂₃₇
unambiguous semidirect product	C₇₉⋊D₁₂₃

Splittings

Fitting, socle, derived, Hall	C₃×C₄₁×C₇₉ ⋊ C₂
Hall	C₄₁×C₇₉ ⋊ D₃
Hall	C₃×C₇₉ ⋊ D₄₁
Hall	C₃×C₄₁ ⋊ D₇₉
Sylow	C₇₉ ⋊ D₁₂₃
Sylow	C₄₁ ⋊ D₂₃₇
Sylow	C₃ ⋊ D₃₂₃₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₄₁×C₇₉	C₂
LCS	G ▷ C₃×C₄₁×C₇₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 9717

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);(2,3)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)

Polycyclic presentation:

2,3,41,79|b^a=b^2,c^a=c^40,d^a=d^78

Conjugacy classes & character table

4860 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉₁₅₁
maximal quotient of	D₂₉₁₅₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19434, 9);

D9717 #34085487