Dic₄₅₂₆ #72216004

order	18104
automorphism	Aut(G) order 39104640
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₅₂₆

Splittings

Hall	C₃₁×C₇₃ ⋊ Q₈
Sylow	C₇₃ ⋊ Dic₆₂
Sylow	C₃₁ ⋊ Dic₁₄₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₂₆₃
commutator, socle, prime radical, cosocle	C₂×C₃₁×C₇₃	C₂²
Fitting	C₄×C₃₁×C₇₃	C₂
hypercenter	C₄	D₂₂₆₃
hypocenter	C₃₁×C₇₃	Q₈
LCS	G ▷ C₂×C₃₁×C₇₃ ▷ C₃₁×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 112
transitive degree	μ_t(G) = 18104

A minimal faithful permutation representation — 2 generators on 112 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)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112);(1,5,3,7)(2,6,4,8)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(41,112)(42,111)(43,110)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2^3,31,73|a^2=c,b^2=c,b^a=bc,d^a=d^30,e^a=e^72

Conjugacy classes & character table

4529 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₀₅₂, Dic₁₃₅₇₈
maximal normal subgroup of	Dic₉₀₅₂
maximal quotient of	Dic₁₃₅₇₈

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 31, 73]);;
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.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4526 #72216004