Dic₆₄₇₄ #72236124

order	25896
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₄₇₄

Splittings

Hall	C₃×C₁₃×C₈₃ ⋊ Q₈
Hall	C₁₃×C₈₃ ⋊ Dic₆
Hall	C₃×C₈₃ ⋊ Dic₂₆
Sylow	C₈₃ ⋊ Dic₇₈
Hall	C₃×C₁₃ ⋊ Dic₁₆₆
Sylow	C₁₃ ⋊ Dic₄₉₈
Sylow	C₃ ⋊ Dic₂₁₅₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₂₃₇
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₃×C₈₃	C₂²
Fitting	C₃×C₄×C₁₃×C₈₃	C₂
hypercenter	C₄	D₃₂₃₇
hypocenter	C₃×C₁₃×C₈₃	Q₈
LCS	G ▷ C₂×C₃×C₁₃×C₈₃ ▷ C₃×C₁₃×C₈₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 25896

A minimal faithful permutation representation — 2 generators on 107 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,5,3,7)(2,6,4,8)(10,11)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^3,3,13,83|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^12,f^a=f^82

Conjugacy classes & character table

6477 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₉₄₈
maximal normal subgroup of	Dic₁₂₉₄₈

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, 83]);;
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.6^82);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6474 #72236124