D₂₇₁₄₄ #72290474

order	54288
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₁₄₄

Splittings

Fitting	C₈×C₉×C₁₃×C₂₉ ⋊ C₂
Hall	C₉×C₁₃×C₂₉ ⋊ D₈
Hall	C₁₃×C₂₉ ⋊ D₇₂
Hall	C₉×C₂₉ ⋊ D₁₀₄
Hall	C₉×C₁₃ ⋊ D₂₃₂
Sylow	C₂₉ ⋊ D₉₃₆
Sylow	C₁₃ ⋊ D₂₀₈₈
Sylow	C₉ ⋊ D₃₀₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₃₅₇₂
commutator, cosocle	C₄×C₉×C₁₃×C₂₉	C₂²
Frattini	C₃×C₄	C₂×D₁₁₃₁
Fitting	C₈×C₉×C₁₃×C₂₉	C₂
socle	C₂×C₃×C₁₃×C₂₉	D₁₂
hypercenter	C₈	D₃₃₉₃
hypocenter	C₉×C₁₃×C₂₉	D₈
LCS	G ▷ C₄×C₉×C₁₃×C₂₉ ▷ C₂×C₉×C₁₃×C₂₉ ▷ C₉×C₁₃×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 27144

A minimal faithful permutation representation — 2 generators on 59 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);(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^4,3,13,29,3|b^2=cd,c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^12,g^a=g^28,h^a=h^2

Conjugacy classes & character table

13575 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, 3, 13, 29, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.8);;
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.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27144 #72290474