D₁₄₁₀₀ #72235657

order	28200
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₁₀₀

Splittings

Fitting	C₃×C₄×C₂₅×C₄₇ ⋊ C₂
Hall	C₃×C₂₅×C₄₇ ⋊ D₄
Hall	C₂₅×C₄₇ ⋊ D₁₂
Hall	C₃×C₄₇ ⋊ D₁₀₀
Hall	C₃×C₂₅ ⋊ D₁₈₈
Sylow	C₄₇ ⋊ D₃₀₀
Sylow	C₂₅ ⋊ D₅₆₄
Sylow	C₃ ⋊ D₄₇₀₀

Subgroups

	subgroup	G/H
center	C₂	C₂×D₃₅₂₅
commutator, cosocle	C₂×C₃×C₂₅×C₄₇	C₂²
Frattini	C₂×C₅	C₂×D₇₀₅
Fitting	C₃×C₄×C₂₅×C₄₇	C₂
socle	C₂×C₃×C₅×C₄₇	C₂×D₅
hypercenter	C₄	D₃₅₂₅
hypocenter	C₃×C₂₅×C₄₇	D₄
LCS	G ▷ C₂×C₃×C₂₅×C₄₇ ▷ C₃×C₂₅×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 14100

A minimal faithful permutation representation — 2 generators on 79 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);(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)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2^3,3,5,47,5|b^2=c,e^5=g,b^a=bc,d^a=d^2,e^a=e^4g^4,f^a=f^46,g^a=g^4

Conjugacy classes & character table

7053 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₂₀₀
maximal normal subgroup of	D₂₈₂₀₀
maximal quotient of	Dic₁₄₁₀₀, D₂₈₂₀₀

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, 47, 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.6^46);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14100 #72235657