Dic₅₀₇₄ #72216656

order	20296
automorphism	Aut(G) order 49441056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₀₇₄

Splittings

Hall	C₄₃×C₅₉ ⋊ Q₈
Sylow	C₅₉ ⋊ Dic₈₆
Sylow	C₄₃ ⋊ Dic₁₁₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₅₃₇
commutator, socle, prime radical, cosocle	C₂×C₄₃×C₅₉	C₂²
Fitting	C₄×C₄₃×C₅₉	C₂
hypercenter	C₄	D₂₅₃₇
hypocenter	C₄₃×C₅₉	Q₈
LCS	G ▷ C₂×C₄₃×C₅₉ ▷ C₄₃×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 110
transitive degree	μ_t(G) = 20296

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

Polycyclic presentation:

2^3,43,59|a^2=c,b^2=c,b^a=bc,d^a=d^42,e^a=e^58

Conjugacy classes & character table

5077 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₁₄₈, Dic₁₅₂₂₂
maximal normal subgroup of	Dic₁₀₁₄₈
maximal quotient of	Dic₁₅₂₂₂

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, 43, 59]);;
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.4^42);;
gap> SetConjugate(coll, 5, 1, F.5^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5074 #72216656