Dic₆₁₆₂ #72224326

order	24648
automorphism	Aut(G) order 46141056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₁₆₂

Splittings

Hall	C₃×C₁₃×C₇₉ ⋊ Q₈
Hall	C₁₃×C₇₉ ⋊ Dic₆
Hall	C₃×C₇₉ ⋊ Dic₂₆
Sylow	C₇₉ ⋊ Dic₇₈
Hall	C₃×C₁₃ ⋊ Dic₁₅₈
Sylow	C₁₃ ⋊ Dic₄₇₄
Sylow	C₃ ⋊ Dic₂₀₅₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₀₈₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₃×C₇₉	C₂²
Fitting	C₃×C₄×C₁₃×C₇₉	C₂
hypercenter	C₄	D₃₀₈₁
hypocenter	C₃×C₁₃×C₇₉	Q₈
LCS	G ▷ C₂×C₃×C₁₃×C₇₉ ▷ C₃×C₁₃×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 24648

A minimal faithful permutation representation — 2 generators on 103 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,5,3,7)(2,6,4,8)(10,11)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2^3,3,13,79|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^12,f^a=f^78

Conjugacy classes & character table

6165 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₃₂₄
maximal normal subgroup of	Dic₁₂₃₂₄

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, 13, 79]);;
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.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6162 #72224326