Dic₅₁₇₀ #72216865

order	20680
automorphism	Aut(G) order 38051200
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₁₇₀

Splittings

Hall	C₅×C₁₁×C₄₇ ⋊ Q₈
Hall	C₁₁×C₄₇ ⋊ Dic₁₀
Hall	C₅×C₄₇ ⋊ Dic₂₂
Hall	C₅×C₁₁ ⋊ Dic₉₄
Sylow	C₄₇ ⋊ Dic₁₁₀
Sylow	C₁₁ ⋊ Dic₄₇₀
Sylow	C₅ ⋊ Dic₁₀₃₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₅₈₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₄₇	C₂²
Fitting	C₄×C₅×C₁₁×C₄₇	C₂
hypercenter	C₄	D₂₅₈₅
hypocenter	C₅×C₁₁×C₄₇	Q₈
LCS	G ▷ C₂×C₅×C₁₁×C₄₇ ▷ C₅×C₁₁×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 20680

A minimal faithful permutation representation — 2 generators on 71 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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,24)(16,23)(17,22)(18,21)(19,20)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2^3,5,11,47|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^10,f^a=f^46

Conjugacy classes & character table

5173 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₃₄₀, Dic₁₅₅₁₀
maximal normal subgroup of	Dic₁₀₃₄₀
maximal quotient of	Dic₁₅₅₁₀

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, 5, 11, 47]);;
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.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5170 #72216865