Dic₁₅₂₅₇ #72292943

order	61028
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₂₅₇

Splittings

derived, Hall	C₁₁×C₁₉×C₇₃ ⋊ C₄
Hall	C₁₉×C₇₃ ⋊ Dic₁₁
Hall	C₁₁×C₇₃ ⋊ Dic₁₉
Hall	C₁₁×C₁₉ ⋊ Dic₇₃
Sylow	C₇₃ ⋊ Dic₂₀₉
Sylow	C₁₉ ⋊ Dic₈₀₃
Sylow	C₁₁ ⋊ Dic₁₃₈₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₂₅₇
commutator, hypocenter	C₁₁×C₁₉×C₇₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₉×C₇₃	C₂
LCS	G ▷ C₁₁×C₁₉×C₇₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 61028

A minimal faithful permutation representation — 2 generators on 107 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15)(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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^2,11,19,73|a^2=b,c^a=c^10,d^a=d^18,e^a=e^72

Conjugacy classes & character table

15260 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");;
gap> coll := SingleCollector(F, [2, 2, 11, 19, 73]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15257 #72292943