Dic₆₃₆₄ #72236087

order	25456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₃₆₄

Splittings

Hall	C₃₇×C₄₃ ⋊ Q₁₆
Sylow	C₄₃ ⋊ Dic₁₄₈
Sylow	C₃₇ ⋊ Dic₁₇₂

Subgroups

	subgroup	G/H
center	C₂	D₆₃₆₄
commutator, cosocle	C₄×C₃₇×C₄₃	C₂²
Frattini	C₄	C₂×D₁₅₉₁
Fitting	C₈×C₃₇×C₄₃	C₂
socle, prime radical	C₂×C₃₇×C₄₃	D₄
hypercenter	C₈	D₁₅₉₁
hypocenter	C₃₇×C₄₃	Q₁₆
LCS	G ▷ C₄×C₃₇×C₄₃ ▷ C₂×C₃₇×C₄₃ ▷ C₃₇×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 25456

A minimal faithful permutation representation — 2 generators on 96 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^4,37,43|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^36,f^a=f^42

Conjugacy classes & character table

6367 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₇₂₈
maximal normal subgroup of	Dic₁₂₇₂₈

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, 2, 37, 43]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^36);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6364 #72236087