D₆₆₃₆ #72212295

order	13272
automorphism	Aut(G) order 12422592
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₆₃₆

Splittings

Fitting	C₃×C₄×C₇×C₇₉ ⋊ C₂
Hall	C₃×C₇×C₇₉ ⋊ D₄
Hall	C₇×C₇₉ ⋊ D₁₂
Hall	C₃×C₇₉ ⋊ D₂₈
Sylow	C₇₉ ⋊ D₈₄
Hall	C₃×C₇ ⋊ D₃₁₆
Sylow	C₇ ⋊ D₉₄₈
Sylow	C₃ ⋊ D₂₂₁₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₆₅₉
commutator, socle, cosocle	C₂×C₃×C₇×C₇₉	C₂²
Fitting	C₃×C₄×C₇×C₇₉	C₂
hypercenter	C₄	D₁₆₅₉
hypocenter	C₃×C₇×C₇₉	D₄
LCS	G ▷ C₂×C₃×C₇×C₇₉ ▷ C₃×C₇×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 6636

A minimal faithful permutation representation — 2 generators on 93 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);(2,4)(6,7)(9,14)(10,13)(11,12)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2^3,3,7,79|b^2=c,b^a=bc,d^a=d^2,e^a=e^6,f^a=f^78

Conjugacy classes & character table

3321 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₂₇₂, D₁₉₉₀₈
maximal normal subgroup of	D₁₃₂₇₂
maximal quotient of	D₁₃₂₇₂, Dic₆₆₃₆, D₁₉₉₀₈

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, 2, 2, 3, 7, 79]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6636 #72212295