Dic₁₄₃₀₈ #72290907

order	57232
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₃₀₈

Splittings

Hall	C₄₉×C₇₃ ⋊ Q₁₆
Sylow	C₇₃ ⋊ Dic₁₉₆
Sylow	C₄₉ ⋊ Dic₂₉₂

Subgroups

	subgroup	G/H
center	C₂	D₁₄₃₀₈
commutator, cosocle	C₄×C₄₉×C₇₃	C₂²
Frattini	C₄×C₇	C₂×D₅₁₁
Fitting	C₈×C₄₉×C₇₃	C₂
socle, prime radical	C₂×C₇×C₇₃	D₂₈
hypercenter	C₈	D₃₅₇₇
hypocenter	C₄₉×C₇₃	Q₁₆
LCS	G ▷ C₄×C₄₉×C₇₃ ▷ C₂×C₄₉×C₇₃ ▷ C₄₉×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 138
transitive degree	μ_t(G) = 57232

A minimal faithful permutation representation — 2 generators on 138 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)(66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)

Polycyclic presentation:

2^4,7,73,7|a^2=d,b^2=cd,c^2=d,e^7=g,b^a=bc,c^a=cd,e^a=e^6g^6,f^a=f^72,g^a=g^6

Conjugacy classes & character table

14311 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 7, 73, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^72);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14308 #72290907