Dic₆₄₂₀ #72236091

order	25680
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₄₂₀

Splittings

Hall	C₃×C₅×C₁₀₇ ⋊ Q₁₆
Hall	C₅×C₁₀₇ ⋊ Dic₁₂
Hall	C₃×C₁₀₇ ⋊ Dic₂₀
Sylow	C₁₀₇ ⋊ Dic₆₀
Hall	C₃×C₅ ⋊ Dic₄₂₈
Sylow	C₅ ⋊ Dic₁₂₈₄
Sylow	C₃ ⋊ Dic₂₁₄₀

Subgroups

	subgroup	G/H
center	C₂	D₆₄₂₀
commutator, cosocle	C₃×C₄×C₅×C₁₀₇	C₂²
Frattini	C₄	C₂×D₁₆₀₅
Fitting	C₃×C₅×C₈×C₁₀₇	C₂
socle, prime radical	C₂×C₃×C₅×C₁₀₇	D₄
hypercenter	C₈	D₁₆₀₅
hypocenter	C₃×C₅×C₁₀₇	Q₁₆
LCS	G ▷ C₃×C₄×C₅×C₁₀₇ ▷ C₂×C₃×C₅×C₁₀₇ ▷ C₃×C₅×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 25680

A minimal faithful permutation representation — 2 generators on 131 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,24)(22,23)(26,131)(27,130)(28,129)(29,128)(30,127)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,120)(38,119)(39,118)(40,117)(41,116)(42,115)(43,114)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2^4,3,5,107|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,g^a=g^106

Conjugacy classes & character table

6423 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₈₄₀
maximal normal subgroup of	Dic₁₂₈₄₀

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, 5, 107]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
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);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6420 #72236091