Dic₁₂₀₁₅ #72280579

order	48060
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₀₁₅

Splittings

derived, Hall	C₅×C₂₇×C₈₉ ⋊ C₄
Hall	C₂₇×C₈₉ ⋊ Dic₅
Hall	C₅×C₈₉ ⋊ Dic₂₇
Hall	C₅×C₂₇ ⋊ Dic₈₉
Sylow	C₈₉ ⋊ Dic₁₃₅
Sylow	C₂₇ ⋊ Dic₄₄₅
Sylow	C₅ ⋊ Dic₂₄₀₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₂₀₁₅
commutator, hypocenter	C₅×C₂₇×C₈₉	C₄
Frattini	C₂×C₉	D₁₃₃₅
Fitting, cosocle	C₂×C₅×C₂₇×C₈₉	C₂
socle, prime radical	C₂×C₃×C₅×C₈₉	D₉
LCS	G ▷ C₅×C₂₇×C₈₉
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 48060

A minimal faithful permutation representation — 2 generators on 125 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)(32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,36)(34,35)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^2,3,5,89,3^2|a^2=b,c^3=f,f^3=g,c^a=c^2f^2g^2,d^a=d^4,e^a=e^88,f^a=f^2g^2,g^a=g^2

Conjugacy classes & character table

12018 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");;
gap> coll := SingleCollector(F, [2, 2, 3, 5, 89, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.6^2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^88);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12015 #72280579