D₁₄₆₀₈ #72235866

order	29216
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₆₀₈

Splittings

Fitting	C₁₁×C₁₆×C₈₃ ⋊ C₂
Hall	C₁₁×C₈₃ ⋊ D₁₆
Sylow	C₈₃ ⋊ D₁₇₆
Sylow	C₁₁ ⋊ D₁₃₂₈

Subgroups

	subgroup	G/H
center	C₂	D₇₃₀₄
commutator, cosocle	C₈×C₁₁×C₈₃	C₂²
Frattini	C₈	C₂×D₉₁₃
Fitting	C₁₁×C₁₆×C₈₃	C₂
socle	C₂×C₁₁×C₈₃	D₈
hypercenter	C₁₆	D₉₁₃
hypocenter	C₁₁×C₈₃	D₁₆
LCS	G ▷ C₈×C₁₁×C₈₃ ▷ C₄×C₁₁×C₈₃ ▷ C₂×C₁₁×C₈₃ ▷ C₁₁×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 110
transitive degree	μ_t(G) = 14608

A minimal faithful permutation representation — 2 generators on 110 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2^5,11,83|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^10,g^a=g^82

Conjugacy classes & character table

7307 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉₂₁₆
maximal normal subgroup of	D₂₉₂₁₆
maximal quotient of	Dic₁₄₆₀₈, D₂₉₂₁₆

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 11, 83]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14608 #72235866