Dic₇₄₈₂ #72236601

order	29928
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₄₈₂

Splittings

Hall	C₃×C₂₉×C₄₃ ⋊ Q₈
Hall	C₂₉×C₄₃ ⋊ Dic₆
Hall	C₃×C₄₃ ⋊ Dic₅₈
Hall	C₃×C₂₉ ⋊ Dic₈₆
Sylow	C₄₃ ⋊ Dic₁₇₄
Sylow	C₂₉ ⋊ Dic₂₅₈
Sylow	C₃ ⋊ Dic₂₄₉₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₇₄₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₂₉×C₄₃	C₂²
Fitting	C₃×C₄×C₂₉×C₄₃	C₂
hypercenter	C₄	D₃₇₄₁
hypocenter	C₃×C₂₉×C₄₃	Q₈
LCS	G ▷ C₂×C₃×C₂₉×C₄₃ ▷ C₃×C₂₉×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 29928

A minimal faithful permutation representation — 2 generators on 83 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);(1,5,3,7)(2,6,4,8)(10,11)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(42,83)(43,82)(44,81)(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:

2^3,3,29,43|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^28,f^a=f^42

Conjugacy classes & character table

7485 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₉₆₄
maximal normal subgroup of	Dic₁₄₉₆₄

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, 29, 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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7482 #72236601