Dic₆₅₃₆ #72236150

order	26144
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₅₃₆

Splittings

Hall	C₁₉×C₄₃ ⋊ Q₃₂
Sylow	C₄₃ ⋊ Dic₁₅₂
Sylow	C₁₉ ⋊ Dic₃₄₄

Subgroups

	subgroup	G/H
center	C₂	D₆₅₃₆
commutator, cosocle	C₈×C₁₉×C₄₃	C₂²
Frattini	C₈	C₂×D₈₁₇
Fitting	C₁₆×C₁₉×C₄₃	C₂
socle, prime radical	C₂×C₁₉×C₄₃	D₈
hypercenter	C₁₆	D₈₁₇
hypocenter	C₁₉×C₄₃	Q₃₂
LCS	G ▷ C₈×C₁₉×C₄₃ ▷ C₄×C₁₉×C₄₃ ▷ C₂×C₁₉×C₄₃ ▷ C₁₉×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 94
transitive degree	μ_t(G) = 26144

A minimal faithful permutation representation — 2 generators on 94 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

2^5,19,43|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^18,g^a=g^42

Conjugacy classes & character table

6539 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₀₇₂
maximal normal subgroup of	Dic₁₃₀₇₂

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 19, 43]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6536 #72236150