D₉₁₁₂ #72213946

order	18224
automorphism	Aut(G) order 38489088
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₁₁₂

Splittings

Fitting	C₈×C₁₇×C₆₇ ⋊ C₂
Hall	C₁₇×C₆₇ ⋊ D₈
Sylow	C₆₇ ⋊ D₁₃₆
Sylow	C₁₇ ⋊ D₅₃₆

Subgroups

	subgroup	G/H
center	C₂	D₄₅₅₆
commutator, cosocle	C₄×C₁₇×C₆₇	C₂²
Frattini	C₄	C₂×D₁₁₃₉
Fitting	C₈×C₁₇×C₆₇	C₂
socle	C₂×C₁₇×C₆₇	D₄
hypercenter	C₈	D₁₁₃₉
hypocenter	C₁₇×C₆₇	D₈
LCS	G ▷ C₄×C₁₇×C₆₇ ▷ C₂×C₁₇×C₆₇ ▷ C₁₇×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 9112

A minimal faithful permutation representation — 2 generators on 92 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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92);(2,8)(3,7)(4,6)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,92)(28,91)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Polycyclic presentation:

2^4,17,67|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^16,f^a=f^66

Conjugacy classes & character table

4559 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₈₂₂₄, D₂₇₃₃₆
maximal normal subgroup of	D₁₈₂₂₄
maximal quotient of	Dic₉₁₁₂, D₁₈₂₂₄, D₂₇₃₃₆

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 17, 67]);;
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.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9112 #72213946