D₇₈₀₀ #72212939

order	15600
automorphism	Aut(G) order 14976000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₇₈₀₀

Splittings

Fitting	C₃×C₈×C₁₃×C₂₅ ⋊ C₂
Hall	C₃×C₁₃×C₂₅ ⋊ D₈
Hall	C₁₃×C₂₅ ⋊ D₂₄
Hall	C₃×C₂₅ ⋊ D₁₀₄
Hall	C₃×C₁₃ ⋊ D₂₀₀
Sylow	C₂₅ ⋊ D₃₁₂
Sylow	C₁₃ ⋊ D₆₀₀
Sylow	C₃ ⋊ D₂₆₀₀

Subgroups

	subgroup	G/H
center	C₂	D₃₉₀₀
commutator, cosocle	C₃×C₄×C₁₃×C₂₅	C₂²
Frattini	C₄×C₅	C₂×D₁₉₅
Fitting	C₃×C₈×C₁₃×C₂₅	C₂
socle	C₂×C₃×C₅×C₁₃	D₂₀
hypercenter	C₈	D₉₇₅
hypocenter	C₃×C₁₃×C₂₅	D₈
LCS	G ▷ C₃×C₄×C₁₃×C₂₅ ▷ C₂×C₃×C₁₃×C₂₅ ▷ C₃×C₁₃×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 7800

A minimal faithful permutation representation — 2 generators on 49 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);(2,8)(3,7)(4,6)(10,11)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

2^4,3,5,13,5|b^2=cd,c^2=d,f^5=h,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4h^4,g^a=g^12,h^a=h^4

Conjugacy classes & character table

3903 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₅₆₀₀, D₂₃₄₀₀
maximal normal subgroup of	D₁₅₆₀₀
maximal quotient of	Dic₇₈₀₀, D₁₅₆₀₀, D₂₃₄₀₀

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 5, 13, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4*F.8^4);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D7800 #72212939