D₉₅₆₈ #72214765

order	19136
automorphism	Aut(G) order 40415232
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₅₆₈

Splittings

Fitting	C₁₃×C₂₃×C₃₂ ⋊ C₂
Hall	C₁₃×C₂₃ ⋊ D₃₂
Sylow	C₂₃ ⋊ D₄₁₆
Sylow	C₁₃ ⋊ D₇₃₆

Subgroups

	subgroup	G/H
center	C₂	D₄₇₈₄
commutator, cosocle	C₁₃×C₁₆×C₂₃	C₂²
Frattini	C₁₆	C₂×D₂₉₉
Fitting	C₁₃×C₂₃×C₃₂	C₂
socle	C₂×C₁₃×C₂₃	D₁₆
hypercenter	C₃₂	D₂₉₉
hypocenter	C₁₃×C₂₃	D₃₂
LCS	G ▷ C₁₃×C₁₆×C₂₃ ▷ C₈×C₁₃×C₂₃ ▷ C₄×C₁₃×C₂₃ ▷ C₂×C₁₃×C₂₃ ▷ C₁₃×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 9568

A minimal faithful permutation representation — 2 generators on 68 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2^6,13,23|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^12,h^a=h^22

Conjugacy classes & character table

4787 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉₁₃₆, D₂₈₇₀₄
maximal normal subgroup of	D₁₉₁₃₆
maximal quotient of	Dic₉₅₆₈, D₁₉₁₃₆, D₂₈₇₀₄

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, 2, 2, 13, 23]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.8^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9568 #72214765