Dic₁₉₇₁ #72212349

order	7884
automorphism	Aut(G) order 5108832
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₉₇₁

Splittings

derived, Hall	C₂₇×C₇₃ ⋊ C₄
Sylow	C₇₃ ⋊ Dic₂₇
Sylow	C₂₇ ⋊ Dic₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₉₇₁
commutator, hypocenter	C₂₇×C₇₃	C₄
Frattini	C₂×C₉	D₂₁₉
Fitting, cosocle	C₂×C₂₇×C₇₃	C₂
socle, prime radical	C₂×C₃×C₇₃	D₉
LCS	G ▷ C₂₇×C₇₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 104
transitive degree	μ_t(G) = 7884

A minimal faithful permutation representation — 2 generators on 104 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);(1,3,2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,104)(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:

2^2,3,73,3^2|a^2=b,c^3=e,e^3=f,c^a=c^2e^2f^2,d^a=d^72,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

1974 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₃₉₄₂, Dic₅₉₁₃, Dic₉₈₅₅, …
maximal normal subgroup of	Dic₃₉₄₂
maximal quotient of	Dic₅₉₁₃, Dic₉₈₅₅, Dic₁₃₇₉₇

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 3, 73, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 4, 1, F.4^72);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic1971 #72212349