Dic₁₃₂₇₇ #72288977

order	53108
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₂₇₇

Splittings

derived, Hall	C₁₁×C₁₇×C₇₁ ⋊ C₄
Hall	C₁₇×C₇₁ ⋊ Dic₁₁
Hall	C₁₁×C₇₁ ⋊ Dic₁₇
Hall	C₁₁×C₁₇ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₁₈₇
Sylow	C₁₇ ⋊ Dic₇₈₁
Sylow	C₁₁ ⋊ Dic₁₂₀₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₂₇₇
commutator, hypocenter	C₁₁×C₁₇×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₁₇×C₇₁	C₂
LCS	G ▷ C₁₁×C₁₇×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 53108

A minimal faithful permutation representation — 2 generators on 103 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^2,11,17,71|a^2=b,c^a=c^10,d^a=d^16,e^a=e^70

Conjugacy classes & character table

13280 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, 17, 71]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13277 #72288977