D₁₂₄₀₈ #72224099

order	24816
automorphism	Aut(G) order 45661440
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₄₀₈

Splittings

Fitting	C₃×C₈×C₁₁×C₄₇ ⋊ C₂
Hall	C₃×C₁₁×C₄₇ ⋊ D₈
Hall	C₁₁×C₄₇ ⋊ D₂₄
Hall	C₃×C₄₇ ⋊ D₈₈
Sylow	C₄₇ ⋊ D₂₆₄
Hall	C₃×C₁₁ ⋊ D₃₇₆
Sylow	C₁₁ ⋊ D₁₁₂₈
Sylow	C₃ ⋊ D₄₁₃₆

Subgroups

	subgroup	G/H
center	C₂	D₆₂₀₄
commutator, cosocle	C₃×C₄×C₁₁×C₄₇	C₂²
Frattini	C₄	C₂×D₁₅₅₁
Fitting	C₃×C₈×C₁₁×C₄₇	C₂
socle	C₂×C₃×C₁₁×C₄₇	D₄
hypercenter	C₈	D₁₅₅₁
hypocenter	C₃×C₁₁×C₄₇	D₈
LCS	G ▷ C₃×C₄×C₁₁×C₄₇ ▷ C₂×C₃×C₁₁×C₄₇ ▷ C₃×C₁₁×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 12408

A minimal faithful permutation representation — 2 generators on 69 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);(2,8)(3,7)(4,6)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2^4,3,11,47|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^10,g^a=g^46

Conjugacy classes & character table

6207 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₄₈₁₆
maximal normal subgroup of	D₂₄₈₁₆
maximal quotient of	Dic₁₂₄₀₈, D₂₄₈₁₆

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, 3, 11, 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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D12408 #72224099