Dic₂₁₃₂ #72212475

order	8528
automorphism	Aut(G) order 8186880
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₁₃₂

Splittings

Hall	C₁₃×C₄₁ ⋊ Q₁₆
Sylow	C₄₁ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₁₆₄

Subgroups

	subgroup	G/H
center	C₂	D₂₁₃₂
commutator, cosocle	C₄×C₁₃×C₄₁	C₂²
Frattini	C₄	C₂×D₅₃₃
Fitting	C₈×C₁₃×C₄₁	C₂
socle, prime radical	C₂×C₁₃×C₄₁	D₄
hypercenter	C₈	D₅₃₃
hypocenter	C₁₃×C₄₁	Q₁₆
LCS	G ▷ C₄×C₁₃×C₄₁ ▷ C₂×C₁₃×C₄₁ ▷ C₁₃×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 8528

A minimal faithful permutation representation — 2 generators on 70 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(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);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(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:

2^4,13,41|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^40

Conjugacy classes & character table

2135 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₄₂₆₄, Dic₆₃₉₆, Dic₁₀₆₆₀, …
maximal normal subgroup of	Dic₄₂₆₄
maximal quotient of	Dic₆₃₉₆, Dic₁₀₆₆₀, Dic₁₄₉₂₄

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, 2, 13, 41]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic2132 #72212475