D₂₇₁₂₀ #72290448

order	54240
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₁₂₀

Splittings

Fitting	C₃×C₅×C₁₆×C₁₁₃ ⋊ C₂
Hall	C₃×C₅×C₁₁₃ ⋊ D₁₆
Hall	C₅×C₁₁₃ ⋊ D₄₈
Hall	C₃×C₁₁₃ ⋊ D₈₀
Sylow	C₁₁₃ ⋊ D₂₄₀
Hall	C₃×C₅ ⋊ D₁₈₀₈
Sylow	C₅ ⋊ D₅₄₂₄
Sylow	C₃ ⋊ D₉₀₄₀

Subgroups

	subgroup	G/H
center	C₂	D₁₃₅₆₀
commutator, cosocle	C₃×C₅×C₈×C₁₁₃	C₂²
Frattini	C₈	C₂×D₁₆₉₅
Fitting	C₃×C₅×C₁₆×C₁₁₃	C₂
socle	C₂×C₃×C₅×C₁₁₃	D₈
hypercenter	C₁₆	D₁₆₉₅
hypocenter	C₃×C₅×C₁₁₃	D₁₆
LCS	G ▷ C₃×C₅×C₈×C₁₁₃ ▷ C₃×C₄×C₅×C₁₁₃ ▷ C₂×C₃×C₅×C₁₁₃ ▷ C₃×C₅×C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 27120

A minimal faithful permutation representation — 2 generators on 137 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,24)(22,23)(26,137)(27,136)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^5,3,5,113|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^4,h^a=h^112

Conjugacy classes & character table

13563 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, 5, 113]);;
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.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^112);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27120 #72290448