Dic₄₃₈₆ #72215826

order	17544
automorphism	Aut(G) order 23579136
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₈₆

Splittings

Hall	C₃×C₁₇×C₄₃ ⋊ Q₈
Hall	C₁₇×C₄₃ ⋊ Dic₆
Hall	C₃×C₄₃ ⋊ Dic₃₄
Hall	C₃×C₁₇ ⋊ Dic₈₆
Sylow	C₄₃ ⋊ Dic₁₀₂
Sylow	C₁₇ ⋊ Dic₂₅₈
Sylow	C₃ ⋊ Dic₁₄₆₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₉₃
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₇×C₄₃	C₂²
Fitting	C₃×C₄×C₁₇×C₄₃	C₂
hypercenter	C₄	D₂₁₉₃
hypocenter	C₃×C₁₇×C₄₃	Q₈
LCS	G ▷ C₂×C₃×C₁₇×C₄₃ ▷ C₃×C₁₇×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 17544

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,11)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Polycyclic presentation:

2^3,3,17,43|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^16,f^a=f^42

Conjugacy classes & character table

4389 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₇₂, Dic₁₃₁₅₈
maximal normal subgroup of	Dic₈₇₇₂
maximal quotient of	Dic₁₃₁₅₈

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, 2, 3, 17, 43]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4386 #72215826