D₁₂₈₈ #72210153

order	2576
automorphism	Aut(G) order 680064
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₂₈₈

Splittings

Fitting	C₇×C₈×C₂₃ ⋊ C₂
Hall	C₇×C₂₃ ⋊ D₈
Sylow	C₂₃ ⋊ D₅₆
Sylow	C₇ ⋊ D₁₈₄

Subgroups

	subgroup	G/H
center	C₂	D₆₄₄
commutator, cosocle	C₄×C₇×C₂₃	C₂²
Frattini	C₄	C₂×D₁₆₁
Fitting	C₇×C₈×C₂₃	C₂
socle	C₂×C₇×C₂₃	D₄
hypercenter	C₈	D₁₆₁
hypocenter	C₇×C₂₃	D₈
LCS	G ▷ C₄×C₇×C₂₃ ▷ C₂×C₇×C₂₃ ▷ C₇×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 1288

A minimal faithful permutation representation — 2 generators on 38 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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Polycyclic presentation:

2^4,7,23|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^22

Conjugacy classes & character table

647 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₅₇₆, D₃₈₆₄, D₆₄₄₀, …
maximal normal subgroup of	D₂₅₇₆
maximal quotient of	D₂₅₇₆, D₃₈₆₄, Dic₁₂₈₈, …

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, 7, 23]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D1288 #72210153