D₃₂₇₁₂ #72304268

order	65424
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₇₁₂

Splittings

Fitting	C₃×C₈×C₂₉×C₄₇ ⋊ C₂
Hall	C₃×C₂₉×C₄₇ ⋊ D₈
Hall	C₂₉×C₄₇ ⋊ D₂₄
Hall	C₃×C₄₇ ⋊ D₂₃₂
Hall	C₃×C₂₉ ⋊ D₃₇₆
Sylow	C₄₇ ⋊ D₆₉₆
Sylow	C₂₉ ⋊ D₁₁₂₈
Sylow	C₃ ⋊ D₁₀₉₀₄

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₅₆
commutator, cosocle	C₃×C₄×C₂₉×C₄₇	C₂²
Frattini	C₄	C₂×D₄₀₈₉
Fitting	C₃×C₈×C₂₉×C₄₇	C₂
socle	C₂×C₃×C₂₉×C₄₇	D₄
hypercenter	C₈	D₄₀₈₉
hypocenter	C₃×C₂₉×C₄₇	D₈
LCS	G ▷ C₃×C₄×C₂₉×C₄₇ ▷ C₂×C₃×C₂₉×C₄₇ ▷ C₃×C₂₉×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 32712

A minimal faithful permutation representation — 2 generators on 87 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);(2,8)(3,7)(4,6)(10,11)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2^4,3,29,47|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^28,g^a=g^46

Conjugacy classes & character table

16359 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, 3, 29, 47]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32712 #72304268