Dic₂₄₁₄ #72212733

order	9656
automorphism	Aut(G) order 10814720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₄₁₄

Splittings

Hall	C₁₇×C₇₁ ⋊ Q₈
Sylow	C₇₁ ⋊ Dic₃₄
Sylow	C₁₇ ⋊ Dic₁₄₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₂₀₇
commutator, socle, prime radical, cosocle	C₂×C₁₇×C₇₁	C₂²
Fitting	C₄×C₁₇×C₇₁	C₂
hypercenter	C₄	D₁₂₀₇
hypocenter	C₁₇×C₇₁	Q₈
LCS	G ▷ C₂×C₁₇×C₇₁ ▷ C₁₇×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 9656

A minimal faithful permutation representation — 2 generators on 96 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96);(1,5,3,7)(2,6,4,8)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2^3,17,71|a^2=c,b^2=c,b^a=bc,d^a=d^16,e^a=e^70

Conjugacy classes & character table

2417 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₈₂₈, Dic₇₂₄₂, Dic₁₂₀₇₀
maximal normal subgroup of	Dic₄₈₂₈
maximal quotient of	Dic₇₂₄₂, Dic₁₂₀₇₀

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 17, 71]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2414 #72212733