Dic₁₅₉₅₃ #72297022

order	63812
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₉₅₃

Splittings

derived, Hall	C₇×C₄₃×C₅₃ ⋊ C₄
Hall	C₄₃×C₅₃ ⋊ Dic₇
Hall	C₇×C₅₃ ⋊ Dic₄₃
Hall	C₇×C₄₃ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₃₀₁
Sylow	C₄₃ ⋊ Dic₃₇₁
Sylow	C₇ ⋊ Dic₂₂₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₉₅₃
commutator, hypocenter	C₇×C₄₃×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₄₃×C₅₃	C₂
LCS	G ▷ C₇×C₄₃×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 63812

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)(6,11)(7,10)(8,9)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^2,7,43,53|a^2=b,c^a=c^6,d^a=d^42,e^a=e^52

Conjugacy classes & character table

15956 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, 7, 43, 53]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^42);;
gap> SetConjugate(coll, 5, 1, F.5^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15953 #72297022