Dic₄₀₂₉ #30633071

order	16116
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₂₉
unambiguous semidirect product	C₃⋊Dic₁₃₄₃
unambiguous semidirect product	C₇₉⋊Dic₅₁

Splittings

derived, Hall	C₃×C₁₇×C₇₉ ⋊ C₄
Hall	C₁₇×C₇₉ ⋊ Dic₃
Hall	C₃×C₇₉ ⋊ Dic₁₇
Sylow	C₇₉ ⋊ Dic₅₁
Hall	C₃×C₁₇ ⋊ Dic₇₉
Sylow	C₁₇ ⋊ Dic₂₃₇
Sylow	C₃ ⋊ Dic₁₃₄₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₀₂₉
commutator, hypocenter	C₃×C₁₇×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₇×C₇₉	C₂
LCS	G ▷ C₃×C₁₇×C₇₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 16116

A minimal faithful permutation representation — 2 generators on 103 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(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);(1,3,2,4)(6,7)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(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:

2,3,17,79,2|a^2=e,b^a=b^2,c^a=c^16,d^a=d^78

Conjugacy classes & character table

4032 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₀₅₈
maximal normal subgroup of	Dic₈₀₅₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16116, 9);

Dic4029 #30633071