Dic₂₅₅₆ #72212792

order	10224
automorphism	Aut(G) order 8588160
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₅₅₆

Splittings

Hall	C₉×C₇₁ ⋊ Q₁₆
Sylow	C₇₁ ⋊ Dic₃₆
Sylow	C₉ ⋊ Dic₂₈₄

Subgroups

	subgroup	G/H
center	C₂	D₂₅₅₆
commutator, cosocle	C₄×C₉×C₇₁	C₂²
Frattini	C₃×C₄	C₂×D₂₁₃
Fitting	C₈×C₉×C₇₁	C₂
socle, prime radical	C₂×C₃×C₇₁	D₁₂
hypercenter	C₈	D₆₃₉
hypocenter	C₉×C₇₁	Q₁₆
LCS	G ▷ C₄×C₉×C₇₁ ▷ C₂×C₉×C₇₁ ▷ C₉×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 10224

A minimal faithful permutation representation — 2 generators on 96 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,25)(19,24)(20,23)(21,22)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2^4,3,71,3|a^2=d,b^2=cd,c^2=d,e^3=g,b^a=bc,c^a=cd,e^a=e^2g^2,f^a=f^70,g^a=g^2

Conjugacy classes & character table

2559 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₅₁₁₂, Dic₇₆₆₈, Dic₁₂₇₈₀
maximal normal subgroup of	Dic₅₁₁₂
maximal quotient of	Dic₇₆₆₈, Dic₁₂₇₈₀

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, 2, 2, 2, 3, 71, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2556 #72212792