Dic₃₆₉₂ #72214830

order	14768
automorphism	Aut(G) order 24810240
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₆₉₂

Splittings

Hall	C₁₃×C₇₁ ⋊ Q₁₆
Sylow	C₇₁ ⋊ Dic₅₂
Sylow	C₁₃ ⋊ Dic₂₈₄

Subgroups

	subgroup	G/H
center	C₂	D₃₆₉₂
commutator, cosocle	C₄×C₁₃×C₇₁	C₂²
Frattini	C₄	C₂×D₉₂₃
Fitting	C₈×C₁₃×C₇₁	C₂
socle, prime radical	C₂×C₁₃×C₇₁	D₄
hypercenter	C₈	D₉₂₃
hypocenter	C₁₃×C₇₁	Q₁₆
LCS	G ▷ C₄×C₁₃×C₇₁ ▷ C₂×C₁₃×C₇₁ ▷ C₁₃×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 100
transitive degree	μ_t(G) = 14768

A minimal faithful permutation representation — 2 generators on 100 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100);(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)(31,100)(32,99)(33,98)(34,97)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)

Polycyclic presentation:

2^4,13,71|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^12,f^a=f^70

Conjugacy classes & character table

3695 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₃₈₄, Dic₁₁₀₇₆
maximal normal subgroup of	Dic₇₃₈₄
maximal quotient of	Dic₁₁₀₇₆

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, 71]);;
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.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3692 #72214830