Dic₄₄₅₂ #72215919

order	17808
automorphism	Aut(G) order 22224384
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₄₅₂

Splittings

Hall	C₃×C₇×C₅₃ ⋊ Q₁₆
Hall	C₇×C₅₃ ⋊ Dic₁₂
Hall	C₃×C₅₃ ⋊ Dic₂₈
Sylow	C₅₃ ⋊ Dic₈₄
Hall	C₃×C₇ ⋊ Dic₂₁₂
Sylow	C₇ ⋊ Dic₆₃₆
Sylow	C₃ ⋊ Dic₁₄₈₄

Subgroups

	subgroup	G/H
center	C₂	D₄₄₅₂
commutator, cosocle	C₃×C₄×C₇×C₅₃	C₂²
Frattini	C₄	C₂×D₁₁₁₃
Fitting	C₃×C₇×C₈×C₅₃	C₂
socle, prime radical	C₂×C₃×C₇×C₅₃	D₄
hypercenter	C₈	D₁₁₁₃
hypocenter	C₃×C₇×C₅₃	Q₁₆
LCS	G ▷ C₃×C₄×C₇×C₅₃ ▷ C₂×C₃×C₇×C₅₃ ▷ C₃×C₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 17808

A minimal faithful permutation representation — 2 generators on 79 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,19)(21,26)(22,25)(23,24)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2^4,3,7,53|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^52

Conjugacy classes & character table

4455 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₉₀₄, Dic₁₃₃₅₆
maximal normal subgroup of	Dic₈₉₀₄
maximal quotient of	Dic₁₃₃₅₆

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, 53]);;
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.7^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4452 #72215919