Dic₇₁₉₈ #72236458

order	28792
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₁₉₈

Splittings

Hall	C₅₉×C₆₁ ⋊ Q₈
Sylow	C₆₁ ⋊ Dic₁₁₈
Sylow	C₅₉ ⋊ Dic₁₂₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₅₉₉
commutator, socle, prime radical, cosocle	C₂×C₅₉×C₆₁	C₂²
Fitting	C₄×C₅₉×C₆₁	C₂
hypercenter	C₄	D₃₅₉₉
hypocenter	C₅₉×C₆₁	Q₈
LCS	G ▷ C₂×C₅₉×C₆₁ ▷ C₅₉×C₆₁
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 28792

A minimal faithful permutation representation — 2 generators on 128 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)(68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128);(1,5,3,7)(2,6,4,8)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(69,128)(70,127)(71,126)(72,125)(73,124)(74,123)(75,122)(76,121)(77,120)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)

Polycyclic presentation:

2^3,59,61|a^2=c,b^2=c,b^a=bc,d^a=d^58,e^a=e^60

Conjugacy classes & character table

7201 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₃₉₆
maximal normal subgroup of	Dic₁₄₃₉₆

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 59, 61]);;
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.4^58);;
gap> SetConjugate(coll, 5, 1, F.5^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic7198 #72236458