Dic₁₅₉₃₀ #72296549

order	63720
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₉₃₀

Splittings

Hall	C₅×C₂₇×C₅₉ ⋊ Q₈
Hall	C₂₇×C₅₉ ⋊ Dic₁₀
Hall	C₅×C₅₉ ⋊ Dic₅₄
Hall	C₅×C₂₇ ⋊ Dic₁₁₈
Sylow	C₅₉ ⋊ Dic₂₇₀
Sylow	C₂₇ ⋊ Dic₅₉₀
Sylow	C₅ ⋊ Dic₃₁₈₆

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₉₆₅
commutator, cosocle	C₂×C₅×C₂₇×C₅₉	C₂²
Frattini	C₂×C₉	C₂×D₈₈₅
Fitting	C₄×C₅×C₂₇×C₅₉	C₂
socle, prime radical	C₂×C₃×C₅×C₅₉	C₂×D₉
hypercenter	C₄	D₇₉₆₅
hypocenter	C₅×C₂₇×C₅₉	Q₈
LCS	G ▷ C₂×C₅×C₂₇×C₅₉ ▷ C₅×C₂₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 99
transitive degree	μ_t(G) = 63720

A minimal faithful permutation representation — 2 generators on 99 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);(1,5,3,7)(2,6,4,8)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,40)(38,39)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2^3,3,5,59,3^2|a^2=c,b^2=c,d^3=g,g^3=h,b^a=bc,d^a=d^2g^2h^2,e^a=e^4,f^a=f^58,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

15933 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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 59, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^58);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15930 #72296549