D₁₀₆₆₅ #72215554

order	21330
automorphism	Aut(G) order 59894640
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₀₆₆₅

Splittings

Fitting, derived, Hall	C₅×C₂₇×C₇₉ ⋊ C₂
Hall	C₂₇×C₇₉ ⋊ D₅
Hall	C₅×C₇₉ ⋊ D₂₇
Hall	C₅×C₂₇ ⋊ D₇₉
Sylow	C₇₉ ⋊ D₁₃₅
Sylow	C₂₇ ⋊ D₃₉₅
Sylow	C₅ ⋊ D₂₁₃₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₂₇×C₇₉	C₂
Frattini	C₉	D₁₁₈₅
socle	C₃×C₅×C₇₉	D₉
LCS	G ▷ C₅×C₂₇×C₇₉
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 10665

A minimal faithful permutation representation — 2 generators on 111 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);(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)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2,3,5,79,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^4,d^a=d^78,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

5334 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₀₆₆₅, D₃₁₉₉₅
maximal normal subgroup of	C₂×D₁₀₆₆₅
maximal quotient of	Dic₁₀₆₆₅, C₂×D₁₀₆₆₅, D₃₁₉₉₅

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, 79, 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.4^78);;
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);

D10665 #72215554