Dic₉₁₀ #72211240

order	3640
automorphism	Aut(G) order 1048320
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₁₀

Splittings

Hall	C₅×C₇×C₁₃ ⋊ Q₈
Hall	C₇×C₁₃ ⋊ Dic₁₀
Hall	C₅×C₁₃ ⋊ Dic₁₄
Hall	C₅×C₇ ⋊ Dic₂₆
Sylow	C₁₃ ⋊ Dic₇₀
Sylow	C₇ ⋊ Dic₁₃₀
Sylow	C₅ ⋊ Dic₁₈₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₄₅₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₇×C₁₃	C₂²
Fitting	C₄×C₅×C₇×C₁₃	C₂
hypercenter	C₄	D₄₅₅
hypocenter	C₅×C₇×C₁₃	Q₈
LCS	G ▷ C₂×C₅×C₇×C₁₃ ▷ C₅×C₇×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 3640

A minimal faithful permutation representation — 2 generators on 33 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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,20)(16,19)(17,18)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Polycyclic presentation:

2^3,5,7,13|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^12

Conjugacy classes & character table

913 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₁₈₂₀, Dic₂₇₃₀, Dic₄₅₅₀, …
maximal normal subgroup of	Dic₁₈₂₀
maximal quotient of	Dic₂₇₃₀, Dic₄₅₅₀, Dic₆₃₇₀, …

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, 7, 13]);;
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.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic910 #72211240