D₉₀₄₈ #72213870

order	18096
automorphism	Aut(G) order 24321024
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₀₄₈

Splittings

Fitting	C₃×C₈×C₁₃×C₂₉ ⋊ C₂
Hall	C₃×C₁₃×C₂₉ ⋊ D₈
Hall	C₁₃×C₂₉ ⋊ D₂₄
Hall	C₃×C₂₉ ⋊ D₁₀₄
Hall	C₃×C₁₃ ⋊ D₂₃₂
Sylow	C₂₉ ⋊ D₃₁₂
Sylow	C₁₃ ⋊ D₆₉₆
Sylow	C₃ ⋊ D₃₀₁₆

Subgroups

	subgroup	G/H
center	C₂	D₄₅₂₄
commutator, cosocle	C₃×C₄×C₁₃×C₂₉	C₂²
Frattini	C₄	C₂×D₁₁₃₁
Fitting	C₃×C₈×C₁₃×C₂₉	C₂
socle	C₂×C₃×C₁₃×C₂₉	D₄
hypercenter	C₈	D₁₁₃₁
hypocenter	C₃×C₁₃×C₂₉	D₈
LCS	G ▷ C₃×C₄×C₁₃×C₂₉ ▷ C₂×C₃×C₁₃×C₂₉ ▷ C₃×C₁₃×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 9048

A minimal faithful permutation representation — 2 generators on 53 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);(2,8)(3,7)(4,6)(10,11)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Polycyclic presentation:

2^4,3,13,29|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^12,g^a=g^28

Conjugacy classes & character table

4527 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₈₀₉₆, D₂₇₁₄₄
maximal normal subgroup of	D₁₈₀₉₆
maximal quotient of	Dic₉₀₄₈, D₁₈₀₉₆, D₂₇₁₄₄

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, 13, 29]);;
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.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9048 #72213870