Dic₁₄₁₀ #72211941

order	5640
automorphism	Aut(G) order 2075520
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₁₀

Splittings

Hall	C₃×C₅×C₄₇ ⋊ Q₈
Hall	C₅×C₄₇ ⋊ Dic₆
Hall	C₃×C₄₇ ⋊ Dic₁₀
Sylow	C₄₇ ⋊ Dic₃₀
Hall	C₃×C₅ ⋊ Dic₉₄
Sylow	C₅ ⋊ Dic₂₈₂
Sylow	C₃ ⋊ Dic₄₇₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₀₅
commutator, socle, prime radical, cosocle	C₂×C₃×C₅×C₄₇	C₂²
Fitting	C₃×C₄×C₅×C₄₇	C₂
hypercenter	C₄	D₇₀₅
hypocenter	C₃×C₅×C₄₇	Q₈
LCS	G ▷ C₂×C₃×C₅×C₄₇ ▷ C₃×C₅×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 5640

A minimal faithful permutation representation — 2 generators on 63 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);(1,5,3,7)(2,6,4,8)(10,11)(13,16)(14,15)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

1413 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₈₂₀, Dic₄₂₃₀, Dic₇₀₅₀, …
maximal normal subgroup of	Dic₂₈₂₀
maximal quotient of	Dic₄₂₃₀, Dic₇₀₅₀, Dic₉₈₇₀, …

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, 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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^46);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1410 #72211941