D₁₄₀₀ #72210192

order	2800
automorphism	Aut(G) order 672000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₀₀

Splittings

Fitting	C₇×C₈×C₂₅ ⋊ C₂
Hall	C₇×C₂₅ ⋊ D₈
Sylow	C₂₅ ⋊ D₅₆
Sylow	C₇ ⋊ D₂₀₀

Subgroups

	subgroup	G/H
center	C₂	D₇₀₀
commutator, cosocle	C₄×C₇×C₂₅	C₂²
Frattini	C₄×C₅	C₂×D₃₅
Fitting	C₇×C₈×C₂₅	C₂
socle	C₂×C₅×C₇	D₂₀
hypercenter	C₈	D₁₇₅
hypocenter	C₇×C₂₅	D₈
LCS	G ▷ C₄×C₇×C₂₅ ▷ C₂×C₇×C₂₅ ▷ C₇×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1400

A minimal faithful permutation representation — 2 generators on 40 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);(2,8)(3,7)(4,6)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(35,40)(36,39)(37,38)

Polycyclic presentation:

2^4,5,7,5|b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^6,g^a=g^4

Conjugacy classes & character table

703 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₂₈₀₀, D₄₂₀₀, D₇₀₀₀, …
maximal normal subgroup of	D₂₈₀₀
maximal quotient of	D₂₈₀₀, D₄₂₀₀, Dic₁₄₀₀, …

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

D1400 #72210192