D₁₃₆₀₀ #72235535

order	27200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₆₀₀

Splittings

Fitting	C₁₇×C₂₅×C₃₂ ⋊ C₂
Hall	C₁₇×C₂₅ ⋊ D₃₂
Sylow	C₂₅ ⋊ D₅₄₄
Sylow	C₁₇ ⋊ D₈₀₀

Subgroups

	subgroup	G/H
center	C₂	D₆₈₀₀
commutator, cosocle	C₁₆×C₁₇×C₂₅	C₂²
Frattini	C₅×C₁₆	C₂×D₈₅
Fitting	C₁₇×C₂₅×C₃₂	C₂
socle	C₂×C₅×C₁₇	D₈₀
hypercenter	C₃₂	D₄₂₅
hypocenter	C₁₇×C₂₅	D₃₂
LCS	G ▷ C₁₆×C₁₇×C₂₅ ▷ C₈×C₁₇×C₂₅ ▷ C₄×C₁₇×C₂₅ ▷ C₂×C₁₇×C₂₅ ▷ C₁₇×C₂₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 13600

A minimal faithful permutation representation — 2 generators on 74 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^6,5,17,5|b^2=cd,c^2=de,d^2=ef,e^2=f,g^5=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4i^4,h^a=h^16,i^a=i^4

Conjugacy classes & character table

6803 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₂₀₀
maximal normal subgroup of	D₂₇₂₀₀
maximal quotient of	Dic₁₃₆₀₀, D₂₇₂₀₀

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 5, 17, 5]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.9^4);;
gap> SetConjugate(coll, 8, 1, F.8^16);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13600 #72235535