Dic₈₄₂₈ #72272810

order	33712
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₄₂₈

Splittings

Hall	C₄₃×C₄₉ ⋊ Q₁₆
Sylow	C₄₉ ⋊ Dic₁₇₂
Sylow	C₄₃ ⋊ Dic₁₉₆

Subgroups

	subgroup	G/H
center	C₂	D₈₄₂₈
commutator, cosocle	C₄×C₄₃×C₄₉	C₂²
Frattini	C₄×C₇	C₂×D₃₀₁
Fitting	C₈×C₄₃×C₄₉	C₂
socle, prime radical	C₂×C₇×C₄₃	D₂₈
hypercenter	C₈	D₂₁₀₇
hypocenter	C₄₃×C₄₉	Q₁₆
LCS	G ▷ C₄×C₄₃×C₄₉ ▷ C₂×C₄₃×C₄₉ ▷ C₄₃×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 33712

A minimal faithful permutation representation — 2 generators on 108 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);(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)(67,108)(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:

2^4,7,43,7|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^42,g^a=g^6

Conjugacy classes & character table

8431 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, 43, 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.6^42);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8428 #72272810