Dic₁₄₃₂₆ #72290934

order	57304
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₂₆

Splittings

Hall	C₁₃×C₁₉×C₂₉ ⋊ Q₈
Hall	C₁₉×C₂₉ ⋊ Dic₂₆
Hall	C₁₃×C₂₉ ⋊ Dic₃₈
Hall	C₁₃×C₁₉ ⋊ Dic₅₈
Sylow	C₂₉ ⋊ Dic₄₉₄
Sylow	C₁₉ ⋊ Dic₇₅₄
Sylow	C₁₃ ⋊ Dic₁₁₀₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₁₆₃
commutator, socle, prime radical, cosocle	C₂×C₁₃×C₁₉×C₂₉	C₂²
Fitting	C₄×C₁₃×C₁₉×C₂₉	C₂
hypercenter	C₄	D₇₁₆₃
hypocenter	C₁₃×C₁₉×C₂₉	Q₈
LCS	G ▷ C₂×C₁₃×C₁₉×C₂₉ ▷ C₁₃×C₁₉×C₂₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 57304

A minimal faithful permutation representation — 2 generators on 69 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);(1,5,3,7)(2,6,4,8)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2^3,13,19,29|a^2=c,b^2=c,b^a=bc,d^a=d^12,e^a=e^18,f^a=f^28

Conjugacy classes & character table

14329 conjugacy classes. The character table is too large to display.

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, 13, 19, 29]);;
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.4^12);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14326 #72290934