D₁₁₁₂₀ #72215835

order	22240
automorphism	Aut(G) order 49105920
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₁₂₀

Splittings

Fitting	C₅×C₁₆×C₁₃₉ ⋊ C₂
Hall	C₅×C₁₃₉ ⋊ D₁₆
Sylow	C₁₃₉ ⋊ D₈₀
Sylow	C₅ ⋊ D₂₂₂₄

Subgroups

	subgroup	G/H
center	C₂	D₅₅₆₀
commutator, cosocle	C₅×C₈×C₁₃₉	C₂²
Frattini	C₈	C₂×D₆₉₅
Fitting	C₅×C₁₆×C₁₃₉	C₂
socle	C₂×C₅×C₁₃₉	D₈
hypercenter	C₁₆	D₆₉₅
hypocenter	C₅×C₁₃₉	D₁₆
LCS	G ▷ C₅×C₈×C₁₃₉ ▷ C₄×C₅×C₁₃₉ ▷ C₂×C₅×C₁₃₉ ▷ C₅×C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 160
transitive degree	μ_t(G) = 11120

A minimal faithful permutation representation — 2 generators on 160 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(23,160)(24,159)(25,158)(26,157)(27,156)(28,155)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

2^5,5,139|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,g^a=g^138

Conjugacy classes & character table

5563 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₂₂₄₀
maximal normal subgroup of	D₂₂₂₄₀
maximal quotient of	Dic₁₁₁₂₀, D₂₂₂₄₀

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, 5, 139]);;
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.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^138);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11120 #72215835