D₄₉₅₆ #72211554

order	9912
automorphism	Aut(G) order 6898752
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₉₅₆

Splittings

Fitting	C₃×C₄×C₇×C₅₉ ⋊ C₂
Hall	C₃×C₇×C₅₉ ⋊ D₄
Hall	C₇×C₅₉ ⋊ D₁₂
Hall	C₃×C₅₉ ⋊ D₂₈
Sylow	C₅₉ ⋊ D₈₄
Hall	C₃×C₇ ⋊ D₂₃₆
Sylow	C₇ ⋊ D₇₀₈
Sylow	C₃ ⋊ D₁₆₅₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₂₃₉
commutator, socle, cosocle	C₂×C₃×C₇×C₅₉	C₂²
Fitting	C₃×C₄×C₇×C₅₉	C₂
hypercenter	C₄	D₁₂₃₉
hypocenter	C₃×C₇×C₅₉	D₄
LCS	G ▷ C₂×C₃×C₇×C₅₉ ▷ C₃×C₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 4956

A minimal faithful permutation representation — 2 generators on 73 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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2^3,3,7,59|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^58

Conjugacy classes & character table

2481 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₉₁₂, D₁₄₈₆₈, D₂₄₇₈₀
maximal normal subgroup of	D₉₉₁₂
maximal quotient of	D₉₉₁₂, Dic₄₉₅₆, D₁₄₈₆₈, …

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, 2, 2, 3, 7, 59]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4956 #72211554