Dic₁₄₉₄₆ #72292299

order	59784
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₄₆

Splittings

Hall	C₃×C₄₇×C₅₃ ⋊ Q₈
Hall	C₄₇×C₅₃ ⋊ Dic₆
Hall	C₃×C₅₃ ⋊ Dic₉₄
Hall	C₃×C₄₇ ⋊ Dic₁₀₆
Sylow	C₅₃ ⋊ Dic₂₈₂
Sylow	C₄₇ ⋊ Dic₃₁₈
Sylow	C₃ ⋊ Dic₄₉₈₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₄₇₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₄₇×C₅₃	C₂²
Fitting	C₃×C₄×C₄₇×C₅₃	C₂
hypercenter	C₄	D₇₄₇₃
hypocenter	C₃×C₄₇×C₅₃	Q₈
LCS	G ▷ C₂×C₃×C₄₇×C₅₃ ▷ C₃×C₄₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 59784

A minimal faithful permutation representation — 2 generators on 111 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);(1,5,3,7)(2,6,4,8)(10,11)(13,58)(14,57)(15,56)(16,55)(17,54)(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)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)

Polycyclic presentation:

2^3,3,47,53|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^46,f^a=f^52

Conjugacy classes & character table

14949 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, 3, 47, 53]);;
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.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^46);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14946 #72292299