D₃₂₁₄₄ #72301935

order	64288
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₁₄₄

Splittings

Fitting	C₁₆×C₄₁×C₄₉ ⋊ C₂
Hall	C₄₁×C₄₉ ⋊ D₁₆
Sylow	C₄₉ ⋊ D₆₅₆
Sylow	C₄₁ ⋊ D₇₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₆₀₇₂
commutator, cosocle	C₈×C₄₁×C₄₉	C₂²
Frattini	C₇×C₈	C₂×D₂₈₇
Fitting	C₁₆×C₄₁×C₄₉	C₂
socle	C₂×C₇×C₄₁	D₅₆
hypercenter	C₁₆	D₂₀₀₉
hypocenter	C₄₁×C₄₉	D₁₆
LCS	G ▷ C₈×C₄₁×C₄₉ ▷ C₄×C₄₁×C₄₉ ▷ C₂×C₄₁×C₄₉ ▷ C₄₁×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 106
transitive degree	μ_t(G) = 32144

A minimal faithful permutation representation — 2 generators on 106 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2^5,7,41,7|b^2=cd,c^2=de,d^2=e,f^7=h,b^a=bc,c^a=cd,d^a=de,f^a=f^6h^6,g^a=g^40,h^a=h^6

Conjugacy classes & character table

16075 conjugacy classes. The character table is too large to display.

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 7, 41, 7]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
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.6^6*F.8^6);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32144 #72301935