Dic₁₃₇₂₇ #72289941

order	54908
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₇₂₇

Splittings

derived, Hall	C₇×C₃₇×C₅₃ ⋊ C₄
Hall	C₃₇×C₅₃ ⋊ Dic₇
Hall	C₇×C₅₃ ⋊ Dic₃₇
Hall	C₇×C₃₇ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₂₅₉
Sylow	C₃₇ ⋊ Dic₃₇₁
Sylow	C₇ ⋊ Dic₁₉₆₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₇₂₇
commutator, hypocenter	C₇×C₃₇×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₃₇×C₅₃	C₂
LCS	G ▷ C₇×C₃₇×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 54908

A minimal faithful permutation representation — 2 generators on 101 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2^2,7,37,53|a^2=b,c^a=c^6,d^a=d^36,e^a=e^52

Conjugacy classes & character table

13730 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, 37, 53]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^36);;
gap> SetConjugate(coll, 5, 1, F.5^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13727 #72289941