D₁₁₂₀₅ #72215876

order	22410
automorphism	Aut(G) order 66154320
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₁₂₀₅

Splittings

Fitting, derived, Hall	C₅×C₂₇×C₈₃ ⋊ C₂
Hall	C₂₇×C₈₃ ⋊ D₅
Hall	C₅×C₈₃ ⋊ D₂₇
Hall	C₅×C₂₇ ⋊ D₈₃
Sylow	C₈₃ ⋊ D₁₃₅
Sylow	C₂₇ ⋊ D₄₁₅
Sylow	C₅ ⋊ D₂₂₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₂₇×C₈₃	C₂
Frattini	C₉	D₁₂₄₅
socle	C₃×C₅×C₈₃	D₉
LCS	G ▷ C₅×C₂₇×C₈₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 11205

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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,92)(58,91)(59,90)(60,89)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)

Polycyclic presentation:

2,3,5,83,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^4,d^a=d^82,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

5604 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₁₂₀₅
maximal normal subgroup of	C₂×D₁₁₂₀₅
maximal quotient of	Dic₁₁₂₀₅, C₂×D₁₁₂₀₅

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");;
gap> coll := SingleCollector(F, [2, 3, 5, 83, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^82);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11205 #72215876