Dic₁₄₅₃₉ #72291329

order	58156
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₅₃₉

Splittings

derived, Hall	C₇×C₃₁×C₆₇ ⋊ C₄
Hall	C₃₁×C₆₇ ⋊ Dic₇
Hall	C₇×C₆₇ ⋊ Dic₃₁
Hall	C₇×C₃₁ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₂₁₇
Sylow	C₃₁ ⋊ Dic₄₆₉
Sylow	C₇ ⋊ Dic₂₀₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₅₃₉
commutator, hypocenter	C₇×C₃₁×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₃₁×C₆₇	C₂
LCS	G ▷ C₇×C₃₁×C₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 58156

A minimal faithful permutation representation — 2 generators on 109 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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,3,2,4)(6,11)(7,10)(8,9)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(44,109)(45,108)(46,107)(47,106)(48,105)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,96)(58,95)(59,94)(60,93)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)

Polycyclic presentation:

2^2,7,31,67|a^2=b,c^a=c^6,d^a=d^30,e^a=e^66

Conjugacy classes & character table

14542 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, 31, 67]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14539 #72291329