D₂₉₉₆₀ #72295310

order	59920
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₉₉₆₀

Splittings

Fitting	C₅×C₇×C₈×C₁₀₇ ⋊ C₂
Hall	C₅×C₇×C₁₀₇ ⋊ D₈
Hall	C₇×C₁₀₇ ⋊ D₄₀
Hall	C₅×C₁₀₇ ⋊ D₅₆
Sylow	C₁₀₇ ⋊ D₂₈₀
Hall	C₅×C₇ ⋊ D₈₅₆
Sylow	C₇ ⋊ D₄₂₈₀
Sylow	C₅ ⋊ D₅₉₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₄₉₈₀
commutator, cosocle	C₄×C₅×C₇×C₁₀₇	C₂²
Frattini	C₄	C₂×D₃₇₄₅
Fitting	C₅×C₇×C₈×C₁₀₇	C₂
socle	C₂×C₅×C₇×C₁₀₇	D₄
hypercenter	C₈	D₃₇₄₅
hypocenter	C₅×C₇×C₁₀₇	D₈
LCS	G ▷ C₄×C₅×C₇×C₁₀₇ ▷ C₂×C₅×C₇×C₁₀₇ ▷ C₅×C₇×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 29960

A minimal faithful permutation representation — 2 generators on 127 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);(2,8)(3,7)(4,6)(10,13)(11,12)(15,20)(16,19)(17,18)(22,127)(23,126)(24,125)(25,124)(26,123)(27,122)(28,121)(29,120)(30,119)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Polycyclic presentation:

2^4,5,7,107|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^6,g^a=g^106

Conjugacy classes & character table

14983 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, 5, 7, 107]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29960 #72295310