Dic₁₄₂₂ #72211947

order	5688
automorphism	Aut(G) order 2661984
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₂₂

Splittings

Hall	C₉×C₇₉ ⋊ Q₈
Sylow	C₇₉ ⋊ Dic₁₈
Sylow	C₉ ⋊ Dic₁₅₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₁₁
commutator, cosocle	C₂×C₉×C₇₉	C₂²
Frattini	C₂×C₃	C₂×D₂₃₇
Fitting	C₄×C₉×C₇₉	C₂
socle, prime radical	C₂×C₃×C₇₉	C₂×D₃
hypercenter	C₄	D₇₁₁
hypocenter	C₉×C₇₉	Q₈
LCS	G ▷ C₂×C₉×C₇₉ ▷ C₉×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 5688

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

Polycyclic presentation:

2^3,3,79,3|a^2=c,b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^78,f^a=f^2

Conjugacy classes & character table

1425 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₂₈₄₄, Dic₄₂₆₆, Dic₇₁₁₀, …
maximal normal subgroup of	Dic₂₈₄₄
maximal quotient of	Dic₄₂₆₆, Dic₇₁₁₀, Dic₉₉₅₄, …

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, 3, 79, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.6^2);;
gap> SetConjugate(coll, 5, 1, F.5^78);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1422 #72211947