Dic₂₉ #1179

order	116
isoclinism	isoclinic class of D₂₉
automorphism	Aut(G) = C₂×AGL(1,𝔽₂₉) (order 1624) · Inn(G) = D₂₉ · Out(G) = C₂²×C₇
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₂₉

Splittings

derived, Sylow

C₂₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₉
commutator, hypocenter	C₂₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₂₉	C₂
LCS	G ▷ C₂₉
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 116

A minimal faithful permutation representation — 2 generators on 33 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);(1,3,2,4)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)

Polycyclic presentation:

2^2,29|a^2=b,c^a=c^28

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6599, Dic₅₈, C₄×D₂₉, …
maximal normal subgroup of	#6599, Dic₅₈, C₄×D₂₉, …
maximal quotient of	#6597, C₂×Dic₂₉, C₃×Dic₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(116, 1);

Dic29 #1179