D₇₄₁₆ #72212697

order	14832
automorphism	Aut(G) order 18154368
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₄₁₆

Splittings

Fitting	C₈×C₉×C₁₀₃ ⋊ C₂
Hall	C₉×C₁₀₃ ⋊ D₈
Sylow	C₁₀₃ ⋊ D₇₂
Sylow	C₉ ⋊ D₈₂₄

Subgroups

	subgroup	G/H
center	C₂	D₃₇₀₈
commutator, cosocle	C₄×C₉×C₁₀₃	C₂²
Frattini	C₃×C₄	C₂×D₃₀₉
Fitting	C₈×C₉×C₁₀₃	C₂
socle	C₂×C₃×C₁₀₃	D₁₂
hypercenter	C₈	D₉₂₇
hypocenter	C₉×C₁₀₃	D₈
LCS	G ▷ C₄×C₉×C₁₀₃ ▷ C₂×C₉×C₁₀₃ ▷ C₉×C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 7416

A minimal faithful permutation representation — 2 generators on 120 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,120)(20,119)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(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:

2^4,3,103,3|b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^102,g^a=g^2

Conjugacy classes & character table

3711 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₄₈₃₂, D₂₂₂₄₈
maximal normal subgroup of	D₁₄₈₃₂
maximal quotient of	D₁₄₈₃₂, Dic₇₄₁₆, D₂₂₂₄₈

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, 3, 103, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^102);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7416 #72212697