Dic₁₆₂₈ #72212084

order	6512
automorphism	Aut(G) order 4688640
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₆₂₈

Splittings

Hall	C₁₁×C₃₇ ⋊ Q₁₆
Sylow	C₃₇ ⋊ Dic₄₄
Sylow	C₁₁ ⋊ Dic₁₄₈

Subgroups

	subgroup	G/H
center	C₂	D₁₆₂₈
commutator, cosocle	C₄×C₁₁×C₃₇	C₂²
Frattini	C₄	C₂×D₄₀₇
Fitting	C₈×C₁₁×C₃₇	C₂
socle, prime radical	C₂×C₁₁×C₃₇	D₄
hypercenter	C₈	D₄₀₇
hypocenter	C₁₁×C₃₇	Q₁₆
LCS	G ▷ C₄×C₁₁×C₃₇ ▷ C₂×C₁₁×C₃₇ ▷ C₁₁×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 6512

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,27)(19,26)(20,25)(21,24)(22,23)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)

Polycyclic presentation:

2^4,11,37|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^10,f^a=f^36

Conjugacy classes & character table

1631 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₂₅₆, Dic₄₈₈₄, Dic₈₁₄₀, …
maximal normal subgroup of	Dic₃₂₅₆
maximal quotient of	Dic₄₈₈₄, Dic₈₁₄₀, Dic₁₁₃₉₆

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, 2, 11, 37]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1628 #72212084