Dic₄₄₂₈ #72215907

order	17712
automorphism	Aut(G) order 25505280
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₄₂₈

Splittings

Hall	C₂₇×C₄₁ ⋊ Q₁₆
Sylow	C₄₁ ⋊ Dic₁₀₈
Sylow	C₂₇ ⋊ Dic₁₆₄

Subgroups

	subgroup	G/H
center	C₂	D₄₄₂₈
commutator, cosocle	C₄×C₂₇×C₄₁	C₂²
Frattini	C₄×C₉	C₂×D₁₂₃
Fitting	C₈×C₂₇×C₄₁	C₂
socle, prime radical	C₂×C₃×C₄₁	D₃₆
hypercenter	C₈	D₁₁₀₇
hypocenter	C₂₇×C₄₁	Q₁₆
LCS	G ▷ C₄×C₂₇×C₄₁ ▷ C₂×C₂₇×C₄₁ ▷ C₂₇×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 17712

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

Polycyclic presentation:

2^4,3,41,3^2|a^2=d,b^2=cd,c^2=d,e^3=g,g^3=h,b^a=bc,c^a=cd,e^a=e^2g^2h^2,f^a=f^40,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

4431 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₈₅₆, Dic₁₃₂₈₄
maximal normal subgroup of	Dic₈₈₅₆
maximal quotient of	Dic₁₃₂₈₄

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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 41, 3, 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> SetPower(coll, 7, F.8);;
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*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4428 #72215907