Dic₁₄₂₈ #72211949

order	5712
automorphism	Aut(G) order 2193408
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₂₈

Splittings

Hall	C₃×C₇×C₁₇ ⋊ Q₁₆
Hall	C₇×C₁₇ ⋊ Dic₁₂
Hall	C₃×C₁₇ ⋊ Dic₂₈
Hall	C₃×C₇ ⋊ Dic₆₈
Sylow	C₁₇ ⋊ Dic₈₄
Sylow	C₇ ⋊ Dic₂₀₄
Sylow	C₃ ⋊ Dic₄₇₆

Subgroups

	subgroup	G/H
center	C₂	D₁₄₂₈
commutator, cosocle	C₃×C₄×C₇×C₁₇	C₂²
Frattini	C₄	C₂×D₃₅₇
Fitting	C₃×C₇×C₈×C₁₇	C₂
socle, prime radical	C₂×C₃×C₇×C₁₇	D₄
hypercenter	C₈	D₃₅₇
hypocenter	C₃×C₇×C₁₇	Q₁₆
LCS	G ▷ C₃×C₄×C₇×C₁₇ ▷ C₂×C₃×C₇×C₁₇ ▷ C₃×C₇×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 5712

A minimal faithful permutation representation — 2 generators on 43 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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,26)(22,25)(23,24)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)

Polycyclic presentation:

2^4,3,7,17|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6,g^a=g^16

Conjugacy classes & character table

1431 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₈₅₆, Dic₄₂₈₄, Dic₇₁₄₀, …
maximal normal subgroup of	Dic₂₈₅₆
maximal quotient of	Dic₄₂₈₄, Dic₇₁₄₀, Dic₉₉₉₆, …

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 7, 17]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1428 #72211949