D₉₅₇₆ #72214752

order	19152
automorphism	Aut(G) order 24820992
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₅₇₆

Splittings

Fitting	C₇×C₈×C₉×C₁₉ ⋊ C₂
Hall	C₇×C₉×C₁₉ ⋊ D₈
Hall	C₉×C₁₉ ⋊ D₅₆
Hall	C₇×C₁₉ ⋊ D₇₂
Hall	C₇×C₉ ⋊ D₁₅₂
Sylow	C₁₉ ⋊ D₅₀₄
Sylow	C₉ ⋊ D₁₀₆₄
Sylow	C₇ ⋊ D₁₃₆₈

Subgroups

	subgroup	G/H
center	C₂	D₄₇₈₈
commutator, cosocle	C₄×C₇×C₉×C₁₉	C₂²
Frattini	C₃×C₄	C₂×D₃₉₉
Fitting	C₇×C₈×C₉×C₁₉	C₂
socle	C₂×C₃×C₇×C₁₉	D₁₂
hypercenter	C₈	D₁₁₉₇
hypocenter	C₇×C₉×C₁₉	D₈
LCS	G ▷ C₄×C₇×C₉×C₁₉ ▷ C₂×C₇×C₉×C₁₉ ▷ C₇×C₉×C₁₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 9576

A minimal faithful permutation representation — 2 generators on 43 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,24)(20,23)(21,22)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)

Polycyclic presentation:

2^4,3,7,19,3|b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^6,g^a=g^18,h^a=h^2

Conjugacy classes & character table

4791 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉₁₅₂, D₂₈₇₂₈
maximal normal subgroup of	D₁₉₁₅₂
maximal quotient of	Dic₉₅₇₆, D₁₉₁₅₂, D₂₈₇₂₈

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 7, 19, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
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*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9576 #72214752