Dic₄₆₉₀ #72216136

order	18760
automorphism	Aut(G) order 29715840
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₆₉₀

Splittings

Hall	C₅×C₇×C₆₇ ⋊ Q₈
Hall	C₇×C₆₇ ⋊ Dic₁₀
Hall	C₅×C₆₇ ⋊ Dic₁₄
Sylow	C₆₇ ⋊ Dic₇₀
Hall	C₅×C₇ ⋊ Dic₁₃₄
Sylow	C₇ ⋊ Dic₆₇₀
Sylow	C₅ ⋊ Dic₉₃₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₃₄₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₇×C₆₇	C₂²
Fitting	C₄×C₅×C₇×C₆₇	C₂
hypercenter	C₄	D₂₃₄₅
hypocenter	C₅×C₇×C₆₇	Q₈
LCS	G ▷ C₂×C₅×C₇×C₆₇ ▷ C₅×C₇×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 18760

A minimal faithful permutation representation — 2 generators on 87 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,20)(16,19)(17,18)(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:

2^3,5,7,67|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^66

Conjugacy classes & character table

4693 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₃₈₀, Dic₁₄₀₇₀
maximal normal subgroup of	Dic₉₃₈₀
maximal quotient of	Dic₁₄₀₇₀

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, 5, 7, 67]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4690 #72216136