Dic₁₀₇₀ #72211728

order	4280
automorphism	Aut(G) order 1814720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₇₀

Splittings

Hall	C₅×C₁₀₇ ⋊ Q₈
Sylow	C₁₀₇ ⋊ Dic₁₀
Sylow	C₅ ⋊ Dic₂₁₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₃₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₁₀₇	C₂²
Fitting	C₄×C₅×C₁₀₇	C₂
hypercenter	C₄	D₅₃₅
hypocenter	C₅×C₁₀₇	Q₈
LCS	G ▷ C₂×C₅×C₁₀₇ ▷ C₅×C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 120
transitive degree	μ_t(G) = 4280

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

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,120)(16,119)(17,118)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

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

Conjugacy classes & character table

1073 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₁₄₀, Dic₃₂₁₀, Dic₅₃₅₀, …
maximal normal subgroup of	Dic₂₁₄₀
maximal quotient of	Dic₃₂₁₀, Dic₅₃₅₀, Dic₇₄₉₀, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 5, 107]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^106);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1070 #72211728