D₁₄₉₀₄ #72236014

order	29808
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₉₀₄

Splittings

Fitting	C₈×C₂₃×C₈₁ ⋊ C₂
Hall	C₂₃×C₈₁ ⋊ D₈
Sylow	C₈₁ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₆₄₈

Subgroups

	subgroup	G/H
center	C₂	D₇₄₅₂
commutator, cosocle	C₄×C₂₃×C₈₁	C₂²
Frattini	C₄×C₂₇	C₂×D₆₉
Fitting	C₈×C₂₃×C₈₁	C₂
socle	C₂×C₃×C₂₃	D₁₀₈
hypercenter	C₈	D₁₈₆₃
hypocenter	C₂₃×C₈₁	D₈
LCS	G ▷ C₄×C₂₃×C₈₁ ▷ C₂×C₂₃×C₈₁ ▷ C₂₃×C₈₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 14904

A minimal faithful permutation representation — 2 generators on 112 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);(2,8)(3,7)(4,6)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)

Polycyclic presentation:

2^4,3,23,3^3|b^2=cd,c^2=d,e^3=g,g^3=h,h^3=i,b^a=bc,c^a=cd,e^a=e^2g^2h^2i^2,f^a=f^22,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

7455 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉₈₀₈
maximal normal subgroup of	D₂₉₈₀₈
maximal quotient of	Dic₁₄₉₀₄, D₂₉₈₀₈

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 23, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^22);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14904 #72236014