order	232
isoclinism	isoclinic class of D₂₉
automorphism	Aut(G) = D₄×AGL(1,𝔽₂₉) (order 6496) · Inn(G) = D₂₉ · Out(G) = C₂×C₇×D₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×Dic₂₉

Splittings

derived, Sylow

C₂₉ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₂₉
commutator, hypocenter	C₂₉	C₂×C₄
Frattini	C₂	C₂×D₂₉
Fitting, socle, prime radical	C₂²×C₂₉	C₂
cosocle	C₂×C₂₉	C₂²
LCS	G ▷ C₂₉
UCS	C₁ ◁ C₂²

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 232

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2)(3,4,5,6)(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);(1,2)(7,8,10,12,14,16,18,20,22,24,26,28,30,32,34,35,33,31,29,27,25,23,21,19,17,15,13,11,9)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×Dic₂₉, #90779, C₄⋊Dic₂₉, …
maximal normal subgroup of	C₄×Dic₂₉, #90779, C₄⋊Dic₂₉, …
maximal quotient of	C₂×#6597, #90777, C₄×Dic₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(232, 7);

C2×Dic29 #6603

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×Dic₂₉ #6603