D₃₀₂₅₆ #72295571

order	60512
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₂₅₆

Splittings

Fitting	C₁₆×C₃₁×C₆₁ ⋊ C₂
Hall	C₃₁×C₆₁ ⋊ D₁₆
Sylow	C₆₁ ⋊ D₄₉₆
Sylow	C₃₁ ⋊ D₉₇₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₁₂₈
commutator, cosocle	C₈×C₃₁×C₆₁	C₂²
Frattini	C₈	C₂×D₁₈₉₁
Fitting	C₁₆×C₃₁×C₆₁	C₂
socle	C₂×C₃₁×C₆₁	D₈
hypercenter	C₁₆	D₁₈₉₁
hypocenter	C₃₁×C₆₁	D₁₆
LCS	G ▷ C₈×C₃₁×C₆₁ ▷ C₄×C₃₁×C₆₁ ▷ C₂×C₃₁×C₆₁ ▷ C₃₁×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 30256

A minimal faithful permutation representation — 2 generators on 108 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(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:

2^5,31,61|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^30,g^a=g^60

Conjugacy classes & character table

15131 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 31, 61]);;
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.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30256 #72295571