D₈₆₄₈ #72213562

order	17296
automorphism	Aut(G) order 35007104
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₆₄₈

Splittings

Fitting	C₈×C₂₃×C₄₇ ⋊ C₂
Hall	C₂₃×C₄₇ ⋊ D₈
Sylow	C₄₇ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₃₇₆

Subgroups

	subgroup	G/H
center	C₂	D₄₃₂₄
commutator, cosocle	C₄×C₂₃×C₄₇	C₂²
Frattini	C₄	C₂×D₁₀₈₁
Fitting	C₈×C₂₃×C₄₇	C₂
socle	C₂×C₂₃×C₄₇	D₄
hypercenter	C₈	D₁₀₈₁
hypocenter	C₂₃×C₄₇	D₈
LCS	G ▷ C₄×C₂₃×C₄₇ ▷ C₂×C₂₃×C₄₇ ▷ C₂₃×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 8648

A minimal faithful permutation representation — 2 generators on 78 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);(2,8)(3,7)(4,6)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2^4,23,47|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^46

Conjugacy classes & character table

4327 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₂₉₆, D₂₅₉₄₄
maximal normal subgroup of	D₁₇₂₉₆
maximal quotient of	Dic₈₆₄₈, D₁₇₂₉₆, D₂₅₉₄₄

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, 23, 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.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8648 #72213562