Dic₆₁₆₄ #72224328

order	24656
automorphism	Aut(G) order 71601024
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₆₁₆₄

Splittings

Hall	C₂₃×C₆₇ ⋊ Q₁₆
Sylow	C₆₇ ⋊ Dic₉₂
Sylow	C₂₃ ⋊ Dic₂₆₈

Subgroups

	subgroup	G/H
center	C₂	D₆₁₆₄
commutator, cosocle	C₄×C₂₃×C₆₇	C₂²
Frattini	C₄	C₂×D₁₅₄₁
Fitting	C₈×C₂₃×C₆₇	C₂
socle, prime radical	C₂×C₂₃×C₆₇	D₄
hypercenter	C₈	D₁₅₄₁
hypocenter	C₂₃×C₆₇	Q₁₆
LCS	G ▷ C₄×C₂₃×C₆₇ ▷ C₂×C₂₃×C₆₇ ▷ C₂₃×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 106
transitive degree	μ_t(G) = 24656

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

Polycyclic presentation:

2^4,23,67|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^22,f^a=f^66

Conjugacy classes & character table

6167 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₃₂₈
maximal normal subgroup of	Dic₁₂₃₂₈

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, 23, 67]);;
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.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic6164 #72224328