Dic₁₀₁₆₈ #72274248

order	40672
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₀₁₆₈

Splittings

Hall	C₃₁×C₄₁ ⋊ Q₃₂
Sylow	C₄₁ ⋊ Dic₂₄₈
Sylow	C₃₁ ⋊ Dic₃₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₀₁₆₈
commutator, cosocle	C₈×C₃₁×C₄₁	C₂²
Frattini	C₈	C₂×D₁₂₇₁
Fitting	C₁₆×C₃₁×C₄₁	C₂
socle, prime radical	C₂×C₃₁×C₄₁	D₈
hypercenter	C₁₆	D₁₂₇₁
hypocenter	C₃₁×C₄₁	Q₃₂
LCS	G ▷ C₈×C₃₁×C₄₁ ▷ C₄×C₃₁×C₄₁ ▷ C₂×C₃₁×C₄₁ ▷ C₃₁×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 104
transitive degree	μ_t(G) = 40672

A minimal faithful permutation representation — 2 generators on 104 points:

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

Polycyclic presentation:

2^5,31,41|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^30,g^a=g^40

Conjugacy classes & character table

10171 conjugacy classes. The character table is too large to display.

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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 31, 41]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> SetConjugate(coll, 7, 1, F.7^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic10168 #72274248