Dic₁₃₁₄ #72211876

order	5256
automorphism	Aut(G) order 2270592
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₁₄

Splittings

Hall	C₉×C₇₃ ⋊ Q₈
Sylow	C₇₃ ⋊ Dic₁₈
Sylow	C₉ ⋊ Dic₁₄₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₅₇
commutator, cosocle	C₂×C₉×C₇₃	C₂²
Frattini	C₂×C₃	C₂×D₂₁₉
Fitting	C₄×C₉×C₇₃	C₂
socle, prime radical	C₂×C₃×C₇₃	C₂×D₃
hypercenter	C₄	D₆₅₇
hypocenter	C₉×C₇₃	Q₈
LCS	G ▷ C₂×C₉×C₇₃ ▷ C₉×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 5256

A minimal faithful permutation representation — 2 generators on 90 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90);(1,5,3,7)(2,6,4,8)(10,17)(11,16)(12,15)(13,14)(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:

2^3,3,73,3|a^2=c,b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^72,f^a=f^2

Conjugacy classes & character table

1317 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₆₂₈, Dic₃₉₄₂, Dic₆₅₇₀, …
maximal normal subgroup of	Dic₂₆₂₈
maximal quotient of	Dic₃₉₄₂, Dic₆₅₇₀, Dic₉₁₉₈, …

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, 73, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^72);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1314 #72211876