D₃₂₅₇₁ #72303261

order	65142
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₂₅₇₁

Splittings

Fitting, derived, Hall	C₇×C₉×C₁₁×C₄₇ ⋊ C₂
Hall	C₉×C₁₁×C₄₇ ⋊ D₇
Hall	C₇×C₁₁×C₄₇ ⋊ D₉
Hall	C₇×C₉×C₄₇ ⋊ D₁₁
Hall	C₇×C₉×C₁₁ ⋊ D₄₇
Hall	C₁₁×C₄₇ ⋊ D₆₃
Hall	C₉×C₄₇ ⋊ D₇₇
Hall	C₇×C₄₇ ⋊ D₉₉
Hall	C₉×C₁₁ ⋊ D₃₂₉
Hall	C₇×C₁₁ ⋊ D₄₂₃
Hall	C₇×C₉ ⋊ D₅₁₇
Sylow	C₄₇ ⋊ D₆₉₃
Sylow	C₁₁ ⋊ D₂₉₆₁
Sylow	C₉ ⋊ D₃₆₁₉
Sylow	C₇ ⋊ D₄₆₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₁₁×C₄₇	C₂
Frattini	C₃	D₁₀₈₅₇
socle	C₃×C₇×C₁₁×C₄₇	D₃
LCS	G ▷ C₇×C₉×C₁₁×C₄₇
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 32571

A minimal faithful permutation representation — 2 generators on 74 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);(2,9)(3,8)(4,7)(5,6)(11,16)(12,15)(13,14)(18,27)(19,26)(20,25)(21,24)(22,23)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,3,7,11,47,3|b^3=f,b^a=b^2f^2,c^a=c^6,d^a=d^10,e^a=e^46,f^a=f^2

Conjugacy classes & character table

16287 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");;
gap> coll := SingleCollector(F, [2, 3, 7, 11, 47, 3]);;
gap> SetPower(coll, 2, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^46);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32571 #72303261