D₈₇₀₀ #72213619

order	17400
automorphism	Aut(G) order 19488000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₇₀₀

Splittings

Fitting	C₃×C₄×C₂₅×C₂₉ ⋊ C₂
Hall	C₃×C₂₅×C₂₉ ⋊ D₄
Hall	C₂₅×C₂₉ ⋊ D₁₂
Hall	C₃×C₂₉ ⋊ D₁₀₀
Hall	C₃×C₂₅ ⋊ D₁₁₆
Sylow	C₂₉ ⋊ D₃₀₀
Sylow	C₂₅ ⋊ D₃₄₈
Sylow	C₃ ⋊ D₂₉₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₂₁₇₅
commutator, cosocle	C₂×C₃×C₂₅×C₂₉	C₂²
Frattini	C₂×C₅	C₂×D₄₃₅
Fitting	C₃×C₄×C₂₅×C₂₉	C₂
socle	C₂×C₃×C₅×C₂₉	C₂×D₅
hypercenter	C₄	D₂₁₇₅
hypocenter	C₃×C₂₅×C₂₉	D₄
LCS	G ▷ C₂×C₃×C₂₅×C₂₉ ▷ C₃×C₂₅×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 8700

A minimal faithful permutation representation — 2 generators on 61 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);(2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Polycyclic presentation:

2^3,3,5,29,5|b^2=c,e^5=g,b^a=bc,d^a=d^2,e^a=e^4g^4,f^a=f^28,g^a=g^4

Conjugacy classes & character table

4353 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₄₀₀, D₂₆₁₀₀
maximal normal subgroup of	D₁₇₄₀₀
maximal quotient of	Dic₈₇₀₀, D₁₇₄₀₀, D₂₆₁₀₀

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, 3, 5, 29, 5]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8700 #72213619