Dic₁₃₆₀₁ #72289733

order	54404
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₀₁

Splittings

derived, Hall	C₇×C₂₉×C₆₇ ⋊ C₄
Hall	C₂₉×C₆₇ ⋊ Dic₇
Hall	C₇×C₆₇ ⋊ Dic₂₉
Hall	C₇×C₂₉ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₂₀₃
Sylow	C₂₉ ⋊ Dic₄₆₉
Sylow	C₇ ⋊ Dic₁₉₄₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₆₀₁
commutator, hypocenter	C₇×C₂₉×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₉×C₆₇	C₂
LCS	G ▷ C₇×C₂₉×C₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 54404

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

Polycyclic presentation:

2^2,7,29,67|a^2=b,c^a=c^6,d^a=d^28,e^a=e^66

Conjugacy classes & character table

13604 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, 29, 67]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^6);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13601 #72289733