Dic₃₀₈₂ #72213330

order	12328
automorphism	Aut(G) order 17900256
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₀₈₂

Splittings

Hall	C₂₃×C₆₇ ⋊ Q₈
Sylow	C₆₇ ⋊ Dic₄₆
Sylow	C₂₃ ⋊ Dic₁₃₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₅₄₁
commutator, socle, prime radical, cosocle	C₂×C₂₃×C₆₇	C₂²
Fitting	C₄×C₂₃×C₆₇	C₂
hypercenter	C₄	D₁₅₄₁
hypocenter	C₂₃×C₆₇	Q₈
LCS	G ▷ C₂×C₂₃×C₆₇ ▷ C₂₃×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 98
transitive degree	μ_t(G) = 12328

A minimal faithful permutation representation — 2 generators on 98 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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98);(1,5,3,7)(2,6,4,8)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(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:

2^3,23,67|a^2=c,b^2=c,b^a=bc,d^a=d^22,e^a=e^66

Conjugacy classes & character table

3085 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₆₁₆₄, Dic₉₂₄₆, Dic₁₅₄₁₀
maximal normal subgroup of	Dic₆₁₆₄
maximal quotient of	Dic₉₂₄₆, Dic₁₅₄₁₀

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, 2, 23, 67]);;
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.4^22);;
gap> SetConjugate(coll, 5, 1, F.5^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3082 #72213330