D₆₆₁₅ #72212313

order	13230
automorphism	Aut(G) order 20003760
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₆₆₁₅

Splittings

Fitting, derived, Hall	C₅×C₂₇×C₄₉ ⋊ C₂
Hall	C₂₇×C₄₉ ⋊ D₅
Hall	C₅×C₄₉ ⋊ D₂₇
Hall	C₅×C₂₇ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₁₃₅
Sylow	C₂₇ ⋊ D₂₄₅
Sylow	C₅ ⋊ D₁₃₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₂₇×C₄₉	C₂
Frattini	C₇×C₉	D₁₀₅
socle	C₃×C₅×C₇	D₆₃
LCS	G ▷ C₅×C₂₇×C₄₉
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 6615

A minimal faithful permutation representation — 2 generators on 81 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);(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)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2,3,5,7,3,7,3|b^3=e,d^7=f,e^3=g,b^a=b^2e^2g^2,c^a=c^4,d^a=d^6f^6,e^a=e^2g^2,f^a=f^6,g^a=g^2

Conjugacy classes & character table

3309 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₆₆₁₅, D₁₉₈₄₅
maximal normal subgroup of	C₂×D₆₆₁₅
maximal quotient of	C₂×D₆₆₁₅, Dic₆₆₁₅, D₁₉₈₄₅

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

D6615 #72212313