D₂₉₉₀₄ #72294747

order	59808
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₉₉₀₄

Splittings

Fitting	C₃×C₇×C₁₆×C₈₉ ⋊ C₂
Hall	C₃×C₇×C₈₉ ⋊ D₁₆
Hall	C₇×C₈₉ ⋊ D₄₈
Hall	C₃×C₈₉ ⋊ D₁₁₂
Sylow	C₈₉ ⋊ D₃₃₆
Hall	C₃×C₇ ⋊ D₁₄₂₄
Sylow	C₇ ⋊ D₄₂₇₂
Sylow	C₃ ⋊ D₉₉₆₈

Subgroups

	subgroup	G/H
center	C₂	D₁₄₉₅₂
commutator, cosocle	C₃×C₇×C₈×C₈₉	C₂²
Frattini	C₈	C₂×D₁₈₆₉
Fitting	C₃×C₇×C₁₆×C₈₉	C₂
socle	C₂×C₃×C₇×C₈₉	D₈
hypercenter	C₁₆	D₁₈₆₉
hypocenter	C₃×C₇×C₈₉	D₁₆
LCS	G ▷ C₃×C₇×C₈×C₈₉ ▷ C₃×C₄×C₇×C₈₉ ▷ C₂×C₃×C₇×C₈₉ ▷ C₃×C₇×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 29904

A minimal faithful permutation representation — 2 generators on 115 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,26)(22,25)(23,24)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(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:

2^5,3,7,89|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^6,h^a=h^88

Conjugacy classes & character table

14955 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 7, 89]);;
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.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29904 #72294747