D₁₀₇₀₇ #36339144

order	21414
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₇₀₇
unambiguous semidirect product	C₃⋊D₃₅₆₉
unambiguous semidirect product	C₄₃⋊D₂₄₉
unambiguous semidirect product	C₈₃⋊D₁₂₉

Splittings

Fitting, socle, derived, Hall	C₃×C₄₃×C₈₃ ⋊ C₂
Hall	C₄₃×C₈₃ ⋊ D₃
Hall	C₃×C₈₃ ⋊ D₄₃
Hall	C₃×C₄₃ ⋊ D₈₃
Sylow	C₈₃ ⋊ D₁₂₉
Sylow	C₄₃ ⋊ D₂₄₉
Sylow	C₃ ⋊ D₃₅₆₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₄₃×C₈₃	C₂
LCS	G ▷ C₃×C₄₃×C₈₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 10707

A minimal faithful permutation representation — 2 generators on 129 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);(2,3)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(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:

2,3,43,83|b^a=b^2,c^a=c^42,d^a=d^82

Conjugacy classes & character table

5355 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₂₁₂₁
maximal quotient of	D₃₂₁₂₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21414, 9);

D10707 #36339144