Dic₉₉₆₄ #72274079

order	39856
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₉₆₄

Splittings

Hall	C₄₇×C₅₃ ⋊ Q₁₆
Sylow	C₅₃ ⋊ Dic₁₈₈
Sylow	C₄₇ ⋊ Dic₂₁₂

Subgroups

	subgroup	G/H
center	C₂	D₉₉₆₄
commutator, cosocle	C₄×C₄₇×C₅₃	C₂²
Frattini	C₄	C₂×D₂₄₉₁
Fitting	C₈×C₄₇×C₅₃	C₂
socle, prime radical	C₂×C₄₇×C₅₃	D₄
hypercenter	C₈	D₂₄₉₁
hypocenter	C₄₇×C₅₃	Q₁₆
LCS	G ▷ C₄×C₄₇×C₅₃ ▷ C₂×C₄₇×C₅₃ ▷ C₄₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 39856

A minimal faithful permutation representation — 2 generators on 116 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)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)

Polycyclic presentation:

2^4,47,53|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^46,f^a=f^52

Conjugacy classes & character table

9967 conjugacy classes. The character table is too large to display.

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, 47, 53]);;
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.5^46);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic9964 #72274079