D₃₄₇₂ #72211125

order	6944
automorphism	Aut(G) order 4999680
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₄₇₂

Splittings

Fitting	C₇×C₁₆×C₃₁ ⋊ C₂
Hall	C₇×C₃₁ ⋊ D₁₆
Sylow	C₃₁ ⋊ D₁₁₂
Sylow	C₇ ⋊ D₄₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₇₃₆
commutator, cosocle	C₇×C₈×C₃₁	C₂²
Frattini	C₈	C₂×D₂₁₇
Fitting	C₇×C₁₆×C₃₁	C₂
socle	C₂×C₇×C₃₁	D₈
hypercenter	C₁₆	D₂₁₇
hypocenter	C₇×C₃₁	D₁₆
LCS	G ▷ C₇×C₈×C₃₁ ▷ C₄×C₇×C₃₁ ▷ C₂×C₇×C₃₁ ▷ C₇×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 3472

A minimal faithful permutation representation — 2 generators on 54 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,23)(19,22)(20,21)(25,54)(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:

2^5,7,31|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^6,g^a=g^30

Conjugacy classes & character table

1739 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₆₉₄₄, D₁₀₄₁₆, D₁₇₃₆₀, …
maximal normal subgroup of	D₆₉₄₄
maximal quotient of	D₆₉₄₄, Dic₃₄₇₂, D₁₀₄₁₆, …

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, 7, 31]);;
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.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D3472 #72211125