D₄₈₈₈ #72211522

order	9776
automorphism	Aut(G) order 10792704
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₄₈₈₈

Splittings

Fitting	C₈×C₁₃×C₄₇ ⋊ C₂
Hall	C₁₃×C₄₇ ⋊ D₈
Sylow	C₄₇ ⋊ D₁₀₄
Sylow	C₁₃ ⋊ D₃₇₆

Subgroups

	subgroup	G/H
center	C₂	D₂₄₄₄
commutator, cosocle	C₄×C₁₃×C₄₇	C₂²
Frattini	C₄	C₂×D₆₁₁
Fitting	C₈×C₁₃×C₄₇	C₂
socle	C₂×C₁₃×C₄₇	D₄
hypercenter	C₈	D₆₁₁
hypocenter	C₁₃×C₄₇	D₈
LCS	G ▷ C₄×C₁₃×C₄₇ ▷ C₂×C₁₃×C₄₇ ▷ C₁₃×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 4888

A minimal faithful permutation representation — 2 generators on 68 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);(2,8)(3,7)(4,6)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^4,13,47|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^46

Conjugacy classes & character table

2447 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₉₇₇₆, D₁₄₆₆₄, D₂₄₄₄₀
maximal normal subgroup of	D₉₇₇₆
maximal quotient of	D₉₇₇₆, Dic₄₈₈₈, D₁₄₆₆₄, …

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, 2, 13, 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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D4888 #72211522