Dic₄₇₀₀ #72216150

order	18800
automorphism	Aut(G) order 34592000
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₇₀₀

Splittings

Hall	C₂₅×C₄₇ ⋊ Q₁₆
Sylow	C₄₇ ⋊ Dic₁₀₀
Sylow	C₂₅ ⋊ Dic₁₈₈

Subgroups

	subgroup	G/H
center	C₂	D₄₇₀₀
commutator, cosocle	C₄×C₂₅×C₄₇	C₂²
Frattini	C₄×C₅	C₂×D₂₃₅
Fitting	C₈×C₂₅×C₄₇	C₂
socle, prime radical	C₂×C₅×C₄₇	D₂₀
hypercenter	C₈	D₁₁₇₅
hypocenter	C₂₅×C₄₇	Q₁₆
LCS	G ▷ C₄×C₂₅×C₄₇ ▷ C₂×C₂₅×C₄₇ ▷ C₂₅×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 18800

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

Polycyclic presentation:

2^4,5,47,5|a^2=d,b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^46,g^a=g^4

Conjugacy classes & character table

4703 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₄₀₀, Dic₁₄₁₀₀
maximal normal subgroup of	Dic₉₄₀₀
maximal quotient of	Dic₁₄₁₀₀

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, 5, 47, 5]);;
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.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4700 #72216150