Dic₁₂₄₀₀ #72285532

order	49600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₄₀₀

Splittings

Hall	C₂₅×C₃₁ ⋊ Q₆₄
Sylow	C₃₁ ⋊ Dic₄₀₀
Sylow	C₂₅ ⋊ Dic₄₉₆

Subgroups

	subgroup	G/H
center	C₂	D₁₂₄₀₀
commutator, cosocle	C₁₆×C₂₅×C₃₁	C₂²
Frattini	C₅×C₁₆	C₂×D₁₅₅
Fitting	C₂₅×C₃₁×C₃₂	C₂
socle, prime radical	C₂×C₅×C₃₁	D₈₀
hypercenter	C₃₂	D₇₇₅
hypocenter	C₂₅×C₃₁	Q₆₄
LCS	G ▷ C₁₆×C₂₅×C₃₁ ▷ C₈×C₂₅×C₃₁ ▷ C₄×C₂₅×C₃₁ ▷ C₂×C₂₅×C₃₁ ▷ C₂₅×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 49600

A minimal faithful permutation representation — 2 generators on 120 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(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)(90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)

Polycyclic presentation:

2^6,5,31,5|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,g^5=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^4i^4,h^a=h^30,i^a=i^4

Conjugacy classes & character table

12403 conjugacy classes. The character table is too large to display.

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 5, 31, 5]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^4*F.9^4);;
gap> SetConjugate(coll, 8, 1, F.8^30);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12400 #72285532