Dic₆₄₀₂ #72236083

order	25608
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₄₀₂

Splittings

Hall	C₃×C₁₁×C₉₇ ⋊ Q₈
Hall	C₁₁×C₉₇ ⋊ Dic₆
Hall	C₃×C₉₇ ⋊ Dic₂₂
Sylow	C₉₇ ⋊ Dic₆₆
Hall	C₃×C₁₁ ⋊ Dic₁₉₄
Sylow	C₁₁ ⋊ Dic₅₈₂
Sylow	C₃ ⋊ Dic₂₁₃₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₂₀₁
commutator, socle, prime radical, cosocle	C₂×C₃×C₁₁×C₉₇	C₂²
Fitting	C₃×C₄×C₁₁×C₉₇	C₂
hypercenter	C₄	D₃₂₀₁
hypocenter	C₃×C₁₁×C₉₇	Q₈
LCS	G ▷ C₂×C₃×C₁₁×C₉₇ ▷ C₃×C₁₁×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 25608

A minimal faithful permutation representation — 2 generators on 119 points:

(1,2,3,4)(5,8,7,6)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,5,3,7)(2,6,4,8)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^3,3,11,97|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^10,f^a=f^96

Conjugacy classes & character table

6405 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₈₀₄
maximal normal subgroup of	Dic₁₂₈₀₄

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, 11, 97]);;
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.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6402 #72236083