D₂₅₁₇₅ #72288359

order	50350
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₅₁₇₅

Splittings

Fitting, derived, Hall	C₁₉×C₂₅×C₅₃ ⋊ C₂
Hall	C₂₅×C₅₃ ⋊ D₁₉
Hall	C₁₉×C₅₃ ⋊ D₂₅
Hall	C₁₉×C₂₅ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₄₇₅
Sylow	C₂₅ ⋊ D₁₀₀₇
Sylow	C₁₉ ⋊ D₁₃₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₉×C₂₅×C₅₃	C₂
Frattini	C₅	D₅₀₃₅
socle	C₅×C₁₉×C₅₃	D₅
LCS	G ▷ C₁₉×C₂₅×C₅₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 25175

A minimal faithful permutation representation — 2 generators on 97 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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2,5,19,53,5|b^5=e,b^a=b^4e^4,c^a=c^18,d^a=d^52,e^a=e^4

Conjugacy classes & character table

12589 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");;
gap> coll := SingleCollector(F, [2, 5, 19, 53, 5]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.5^4);;
gap> SetConjugate(coll, 3, 1, F.3^18);;
gap> SetConjugate(coll, 4, 1, F.4^52);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25175 #72288359