D₂₄₄₄₀ #72287910

order	48880
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₄₄₄₀

Splittings

Fitting	C₅×C₈×C₁₃×C₄₇ ⋊ C₂
Hall	C₅×C₁₃×C₄₇ ⋊ D₈
Hall	C₁₃×C₄₇ ⋊ D₄₀
Hall	C₅×C₄₇ ⋊ D₁₀₄
Hall	C₅×C₁₃ ⋊ D₃₇₆
Sylow	C₄₇ ⋊ D₅₂₀
Sylow	C₁₃ ⋊ D₁₈₈₀
Sylow	C₅ ⋊ D₄₈₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₂₂₂₀
commutator, cosocle	C₄×C₅×C₁₃×C₄₇	C₂²
Frattini	C₄	C₂×D₃₀₅₅
Fitting	C₅×C₈×C₁₃×C₄₇	C₂
socle	C₂×C₅×C₁₃×C₄₇	D₄
hypercenter	C₈	D₃₀₅₅
hypocenter	C₅×C₁₃×C₄₇	D₈
LCS	G ▷ C₄×C₅×C₁₃×C₄₇ ▷ C₂×C₅×C₁₃×C₄₇ ▷ C₅×C₁₃×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 24440

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);(2,8)(3,7)(4,6)(10,13)(11,12)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^4,5,13,47|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^12,g^a=g^46

Conjugacy classes & character table

12223 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, 5, 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.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D24440 #72287910