Dic₂₁₃₀ #72212461

order	8520
automorphism	Aut(G) order 4771200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₁₃₀

Splittings

Hall	C₃×C₅×C₇₁ ⋊ Q₈
Hall	C₅×C₇₁ ⋊ Dic₆
Hall	C₃×C₇₁ ⋊ Dic₁₀
Sylow	C₇₁ ⋊ Dic₃₀
Hall	C₃×C₅ ⋊ Dic₁₄₂
Sylow	C₅ ⋊ Dic₄₂₆
Sylow	C₃ ⋊ Dic₇₁₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₀₆₅
commutator, socle, prime radical, cosocle	C₂×C₃×C₅×C₇₁	C₂²
Fitting	C₃×C₄×C₅×C₇₁	C₂
hypercenter	C₄	D₁₀₆₅
hypocenter	C₃×C₅×C₇₁	Q₈
LCS	G ▷ C₂×C₃×C₅×C₇₁ ▷ C₃×C₅×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 8520

A minimal faithful permutation representation — 2 generators on 87 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);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2^3,3,5,71|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^70

Conjugacy classes & character table

2133 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₂₆₀, Dic₆₃₉₀, Dic₁₀₆₅₀, …
maximal normal subgroup of	Dic₄₂₆₀
maximal quotient of	Dic₆₃₉₀, Dic₁₀₆₅₀, Dic₁₄₉₁₀

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 71]);;
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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2130 #72212461