Dic₇₆₂₆ #72272048

order	30504
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₆₂₆

Splittings

Hall	C₃×C₃₁×C₄₁ ⋊ Q₈
Hall	C₃₁×C₄₁ ⋊ Dic₆
Hall	C₃×C₄₁ ⋊ Dic₆₂
Hall	C₃×C₃₁ ⋊ Dic₈₂
Sylow	C₄₁ ⋊ Dic₁₈₆
Sylow	C₃₁ ⋊ Dic₂₄₆
Sylow	C₃ ⋊ Dic₂₅₄₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₈₁₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₃₁×C₄₁	C₂²
Fitting	C₃×C₄×C₃₁×C₄₁	C₂
hypercenter	C₄	D₃₈₁₃
hypocenter	C₃×C₃₁×C₄₁	Q₈
LCS	G ▷ C₂×C₃×C₃₁×C₄₁ ▷ C₃×C₃₁×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 30504

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)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^3,3,31,41|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^30,f^a=f^40

Conjugacy classes & character table

7629 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₂₅₂
maximal normal subgroup of	Dic₁₅₂₅₂

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, 31, 41]);;
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.5^30);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7626 #72272048