Dic₁₃₈₄₆ #72290134

order	55384
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₄₆

Splittings

Hall	C₇×C₂₃×C₄₃ ⋊ Q₈
Hall	C₂₃×C₄₃ ⋊ Dic₁₄
Hall	C₇×C₄₃ ⋊ Dic₄₆
Hall	C₇×C₂₃ ⋊ Dic₈₆
Sylow	C₄₃ ⋊ Dic₃₂₂
Sylow	C₂₃ ⋊ Dic₆₀₂
Sylow	C₇ ⋊ Dic₁₉₇₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₉₂₃
commutator, socle, prime radical, cosocle	C₂×C₇×C₂₃×C₄₃	C₂²
Fitting	C₄×C₇×C₂₃×C₄₃	C₂
hypercenter	C₄	D₆₉₂₃
hypocenter	C₇×C₂₃×C₄₃	Q₈
LCS	G ▷ C₂×C₇×C₂₃×C₄₃ ▷ C₇×C₂₃×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 55384

A minimal faithful permutation representation — 2 generators on 81 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);(1,5,3,7)(2,6,4,8)(10,15)(11,14)(12,13)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(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:

2^3,7,23,43|a^2=c,b^2=c,b^a=bc,d^a=d^6,e^a=e^22,f^a=f^42

Conjugacy classes & character table

13849 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, 7, 23, 43]);;
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.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13846 #72290134