D₁₀₆₆₈ #72215530

order	21336
automorphism	Aut(G) order 32260032
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₀₆₆₈

Splittings

Fitting	C₃×C₄×C₇×C₁₂₇ ⋊ C₂
Hall	C₃×C₇×C₁₂₇ ⋊ D₄
Hall	C₇×C₁₂₇ ⋊ D₁₂
Hall	C₃×C₁₂₇ ⋊ D₂₈
Sylow	C₁₂₇ ⋊ D₈₄
Hall	C₃×C₇ ⋊ D₅₀₈
Sylow	C₇ ⋊ D₁₅₂₄
Sylow	C₃ ⋊ D₃₅₅₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₆₆₇
commutator, socle, cosocle	C₂×C₃×C₇×C₁₂₇	C₂²
Fitting	C₃×C₄×C₇×C₁₂₇	C₂
hypercenter	C₄	D₂₆₆₇
hypocenter	C₃×C₇×C₁₂₇	D₄
LCS	G ▷ C₂×C₃×C₇×C₁₂₇ ▷ C₃×C₇×C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 10668

A minimal faithful permutation representation — 2 generators on 141 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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,141)(17,140)(18,139)(19,138)(20,137)(21,136)(22,135)(23,134)(24,133)(25,132)(26,131)(27,130)(28,129)(29,128)(30,127)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,120)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2^3,3,7,127|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^126

Conjugacy classes & character table

5337 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₁₃₃₆, D₃₂₀₀₄
maximal normal subgroup of	D₂₁₃₃₆
maximal quotient of	Dic₁₀₆₆₈, D₂₁₃₃₆, D₃₂₀₀₄

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 127]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^126);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D10668 #72215530