Dic₉₆₂₈ #72273804

order	38512
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₉₆₂₈

Splittings

Hall	C₂₉×C₈₃ ⋊ Q₁₆
Sylow	C₈₃ ⋊ Dic₁₁₆
Sylow	C₂₉ ⋊ Dic₃₃₂

Subgroups

	subgroup	G/H
center	C₂	D₉₆₂₈
commutator, cosocle	C₄×C₂₉×C₈₃	C₂²
Frattini	C₄	C₂×D₂₄₀₇
Fitting	C₈×C₂₉×C₈₃	C₂
socle, prime radical	C₂×C₂₉×C₈₃	D₄
hypercenter	C₈	D₂₄₀₇
hypocenter	C₂₉×C₈₃	Q₁₆
LCS	G ▷ C₄×C₂₉×C₈₃ ▷ C₂×C₂₉×C₈₃ ▷ C₂₉×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 38512

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

Polycyclic presentation:

2^4,29,83|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^28,f^a=f^82

Conjugacy classes & character table

9631 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, 29, 83]);;
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.5^28);;
gap> SetConjugate(coll, 6, 1, F.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic9628 #72273804