Dic₃₂₅₆ #72213635

order	13024
automorphism	Aut(G) order 18754560
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₂₅₆

Splittings

Hall	C₁₁×C₃₇ ⋊ Q₃₂
Sylow	C₃₇ ⋊ Dic₈₈
Sylow	C₁₁ ⋊ Dic₂₉₆

Subgroups

	subgroup	G/H
center	C₂	D₃₂₅₆
commutator, cosocle	C₈×C₁₁×C₃₇	C₂²
Frattini	C₈	C₂×D₄₀₇
Fitting	C₁₁×C₁₆×C₃₇	C₂
socle, prime radical	C₂×C₁₁×C₃₇	D₈
hypercenter	C₁₆	D₄₀₇
hypocenter	C₁₁×C₃₇	Q₃₂
LCS	G ▷ C₈×C₁₁×C₃₇ ▷ C₄×C₁₁×C₃₇ ▷ C₂×C₁₁×C₃₇ ▷ C₁₁×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 13024

A minimal faithful permutation representation — 2 generators on 80 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,43)(35,42)(36,41)(37,40)(38,39)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2^5,11,37|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^10,g^a=g^36

Conjugacy classes & character table

3259 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₅₁₂, Dic₉₇₆₈, Dic₁₆₂₈₀
maximal normal subgroup of	Dic₆₅₁₂
maximal quotient of	Dic₉₇₆₈, Dic₁₆₂₈₀

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, 2, 11, 37]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^10);;
gap> SetConjugate(coll, 7, 1, F.7^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3256 #72213635