D₇₂₁₆ #72212607

order	14432
automorphism	Aut(G) order 23091200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₂₁₆

Splittings

Fitting	C₁₁×C₁₆×C₄₁ ⋊ C₂
Hall	C₁₁×C₄₁ ⋊ D₁₆
Sylow	C₄₁ ⋊ D₁₇₆
Sylow	C₁₁ ⋊ D₆₅₆

Subgroups

	subgroup	G/H
center	C₂	D₃₆₀₈
commutator, cosocle	C₈×C₁₁×C₄₁	C₂²
Frattini	C₈	C₂×D₄₅₁
Fitting	C₁₁×C₁₆×C₄₁	C₂
socle	C₂×C₁₁×C₄₁	D₈
hypercenter	C₁₆	D₄₅₁
hypocenter	C₁₁×C₄₁	D₁₆
LCS	G ▷ C₈×C₁₁×C₄₁ ▷ C₄×C₁₁×C₄₁ ▷ C₂×C₁₁×C₄₁ ▷ C₁₁×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 7216

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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,27)(19,26)(20,25)(21,24)(22,23)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^5,11,41|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^10,g^a=g^40

Conjugacy classes & character table

3611 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₄₄₃₂, D₂₁₆₄₈
maximal normal subgroup of	D₁₄₄₃₂
maximal quotient of	D₁₄₄₃₂, Dic₇₂₁₆, D₂₁₆₄₈

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, 2, 11, 41]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7216 #72212607