Dic₁₁₉₇₂ #72280096

order	47888
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₁₉₇₂

Splittings

Hall	C₄₁×C₇₃ ⋊ Q₁₆
Sylow	C₇₃ ⋊ Dic₁₆₄
Sylow	C₄₁ ⋊ Dic₂₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₁₉₇₂
commutator, cosocle	C₄×C₄₁×C₇₃	C₂²
Frattini	C₄	C₂×D₂₉₉₃
Fitting	C₈×C₄₁×C₇₃	C₂
socle, prime radical	C₂×C₄₁×C₇₃	D₄
hypercenter	C₈	D₂₉₉₃
hypocenter	C₄₁×C₇₃	Q₁₆
LCS	G ▷ C₄×C₄₁×C₇₃ ▷ C₂×C₄₁×C₇₃ ▷ C₄₁×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 130
transitive degree	μ_t(G) = 47888

A minimal faithful permutation representation — 2 generators on 130 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)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)

Polycyclic presentation:

2^4,41,73|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^40,f^a=f^72

Conjugacy classes & character table

11975 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, 41, 73]);;
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.5^40);;
gap> SetConjugate(coll, 6, 1, F.6^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic11972 #72280096