D₃₁₈₀₀ #72301751

order	63600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₁₈₀₀

Splittings

Fitting	C₃×C₈×C₂₅×C₅₃ ⋊ C₂
Hall	C₃×C₂₅×C₅₃ ⋊ D₈
Hall	C₂₅×C₅₃ ⋊ D₂₄
Hall	C₃×C₅₃ ⋊ D₂₀₀
Hall	C₃×C₂₅ ⋊ D₄₂₄
Sylow	C₅₃ ⋊ D₆₀₀
Sylow	C₂₅ ⋊ D₁₂₇₂
Sylow	C₃ ⋊ D₁₀₆₀₀

Subgroups

	subgroup	G/H
center	C₂	D₁₅₉₀₀
commutator, cosocle	C₃×C₄×C₂₅×C₅₃	C₂²
Frattini	C₄×C₅	C₂×D₇₉₅
Fitting	C₃×C₈×C₂₅×C₅₃	C₂
socle	C₂×C₃×C₅×C₅₃	D₂₀
hypercenter	C₈	D₃₉₇₅
hypocenter	C₃×C₂₅×C₅₃	D₈
LCS	G ▷ C₃×C₄×C₂₅×C₅₃ ▷ C₂×C₃×C₂₅×C₅₃ ▷ C₃×C₂₅×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 31800

A minimal faithful permutation representation — 2 generators on 89 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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(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)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^4,3,5,53,5|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^52,h^a=h^4

Conjugacy classes & character table

15903 conjugacy classes. The character table is too large to display.

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, 53, 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.7^52);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D31800 #72301751