Dic₈₉₀₀ #72273201

order	35600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₈₉₀₀

Splittings

Hall	C₂₅×C₈₉ ⋊ Q₁₆
Sylow	C₈₉ ⋊ Dic₁₀₀
Sylow	C₂₅ ⋊ Dic₃₅₆

Subgroups

	subgroup	G/H
center	C₂	D₈₉₀₀
commutator, cosocle	C₄×C₂₅×C₈₉	C₂²
Frattini	C₄×C₅	C₂×D₄₄₅
Fitting	C₈×C₂₅×C₈₉	C₂
socle, prime radical	C₂×C₅×C₈₉	D₂₀
hypercenter	C₈	D₂₂₂₅
hypocenter	C₂₅×C₈₉	Q₁₆
LCS	G ▷ C₄×C₂₅×C₈₉ ▷ C₂×C₂₅×C₈₉ ▷ C₂₅×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 130
transitive degree	μ_t(G) = 35600

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

Polycyclic presentation:

2^4,5,89,5|a^2=d,b^2=cd,c^2=d,e^5=g,b^a=bc,c^a=cd,e^a=e^4g^4,f^a=f^88,g^a=g^4

Conjugacy classes & character table

8903 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, 5, 89, 5]);;
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.5^4*F.7^4);;
gap> SetConjugate(coll, 6, 1, F.6^88);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic8900 #72273201