Dic₃₁₇₂ #72213484

order	12688
automorphism	Aut(G) order 18270720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₁₇₂

Splittings

Hall	C₁₃×C₆₁ ⋊ Q₁₆
Sylow	C₆₁ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₂₄₄

Subgroups

	subgroup	G/H
center	C₂	D₃₁₇₂
commutator, cosocle	C₄×C₁₃×C₆₁	C₂²
Frattini	C₄	C₂×D₇₉₃
Fitting	C₈×C₁₃×C₆₁	C₂
socle, prime radical	C₂×C₁₃×C₆₁	D₄
hypercenter	C₈	D₇₉₃
hypocenter	C₁₃×C₆₁	Q₁₆
LCS	G ▷ C₄×C₁₃×C₆₁ ▷ C₂×C₁₃×C₆₁ ▷ C₁₃×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 12688

A minimal faithful permutation representation — 2 generators on 90 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2^4,13,61|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^60

Conjugacy classes & character table

3175 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₃₄₄, Dic₉₅₁₆, Dic₁₅₈₆₀
maximal normal subgroup of	Dic₆₃₄₄
maximal quotient of	Dic₉₅₁₆, Dic₁₅₈₆₀

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, 13, 61]);;
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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3172 #72213484