D₁₄₂₈ #72210201

order	2856
automorphism	Aut(G) order 548352
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₂₈

Splittings

Fitting	C₃×C₄×C₇×C₁₇ ⋊ C₂
Hall	C₃×C₇×C₁₇ ⋊ D₄
Hall	C₇×C₁₇ ⋊ D₁₂
Hall	C₃×C₁₇ ⋊ D₂₈
Hall	C₃×C₇ ⋊ D₆₈
Sylow	C₁₇ ⋊ D₈₄
Sylow	C₇ ⋊ D₂₀₄
Sylow	C₃ ⋊ D₄₇₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₅₇
commutator, socle, cosocle	C₂×C₃×C₇×C₁₇	C₂²
Fitting	C₃×C₄×C₇×C₁₇	C₂
hypercenter	C₄	D₃₅₇
hypocenter	C₃×C₇×C₁₇	D₄
LCS	G ▷ C₂×C₃×C₇×C₁₇ ▷ C₃×C₇×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 1428

A minimal faithful permutation representation — 2 generators on 31 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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2^3,3,7,17|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^16

Conjugacy classes & character table

717 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₈₅₆, D₄₂₈₄, D₇₁₄₀, …
maximal normal subgroup of	D₂₈₅₆
maximal quotient of	D₂₈₅₆, D₄₂₈₄, Dic₁₄₂₈, …

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, 3, 7, 17]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1428 #72210201