Dic₁₅₀₁₀ #72292409

order	60040
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₀₁₀

Splittings

Hall	C₅×C₁₉×C₇₉ ⋊ Q₈
Hall	C₁₉×C₇₉ ⋊ Dic₁₀
Hall	C₅×C₇₉ ⋊ Dic₃₈
Hall	C₅×C₁₉ ⋊ Dic₁₅₈
Sylow	C₇₉ ⋊ Dic₁₉₀
Sylow	C₁₉ ⋊ Dic₇₉₀
Sylow	C₅ ⋊ Dic₃₀₀₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₇₅₀₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₁₉×C₇₉	C₂²
Fitting	C₄×C₅×C₁₉×C₇₉	C₂
hypercenter	C₄	D₇₅₀₅
hypocenter	C₅×C₁₉×C₇₉	Q₈
LCS	G ▷ C₂×C₅×C₁₉×C₇₉ ▷ C₅×C₁₉×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 60040

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

Polycyclic presentation:

2^3,5,19,79|a^2=c,b^2=c,b^a=bc,d^a=d^4,e^a=e^18,f^a=f^78

Conjugacy classes & character table

15013 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, 5, 19, 79]);;
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.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15010 #72292409