D₃₀₀₁₆ #72295356

order	60032
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₀₁₆

Splittings

Fitting	C₇×C₆₄×C₆₇ ⋊ C₂
Hall	C₇×C₆₇ ⋊ D₆₄
Sylow	C₆₇ ⋊ D₄₄₈
Sylow	C₇ ⋊ D₄₂₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₀₀₈
commutator, cosocle	C₇×C₃₂×C₆₇	C₂²
Frattini	C₃₂	C₂×D₄₆₉
Fitting	C₇×C₆₄×C₆₇	C₂
socle	C₂×C₇×C₆₇	D₃₂
hypercenter	C₆₄	D₄₆₉
hypocenter	C₇×C₆₇	D₆₄
LCS	G ▷ C₇×C₃₂×C₆₇ ▷ C₇×C₁₆×C₆₇ ▷ C₇×C₈×C₆₇ ▷ C₄×C₇×C₆₇ ▷ C₂×C₇×C₆₇ ▷ C₇×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 138
transitive degree	μ_t(G) = 30016

A minimal faithful permutation representation — 2 generators on 138 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);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,71)(67,70)(68,69)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)

Polycyclic presentation:

2^7,7,67|b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^6,i^a=i^66

Conjugacy classes & character table

15011 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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 7, 67]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7);;
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, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 9, 1, F.9^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30016 #72295356