D₂₅₂₃₂ #72288412

order	50464
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₂₃₂

Splittings

Fitting	C₁₆×C₁₉×C₈₃ ⋊ C₂
Hall	C₁₉×C₈₃ ⋊ D₁₆
Sylow	C₈₃ ⋊ D₃₀₄
Sylow	C₁₉ ⋊ D₁₃₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₂₆₁₆
commutator, cosocle	C₈×C₁₉×C₈₃	C₂²
Frattini	C₈	C₂×D₁₅₇₇
Fitting	C₁₆×C₁₉×C₈₃	C₂
socle	C₂×C₁₉×C₈₃	D₈
hypercenter	C₁₆	D₁₅₇₇
hypocenter	C₁₉×C₈₃	D₁₆
LCS	G ▷ C₈×C₁₉×C₈₃ ▷ C₄×C₁₉×C₈₃ ▷ C₂×C₁₉×C₈₃ ▷ C₁₉×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 118
transitive degree	μ_t(G) = 25232

A minimal faithful permutation representation — 2 generators on 118 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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2^5,19,83|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^18,g^a=g^82

Conjugacy classes & character table

12619 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 19, 83]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25232 #72288412