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C₄×C₂₉ #1180

order	116
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄×C₇ (order 56) · Inn(G) = C₁ · Out(G) = C₂×C₄×C₇
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₂₉)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₁₆
abelian	C₄×C₂₉

Subgroups

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

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 — 1 generator 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)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

center of	M₁₆×C₂₉, #41×C₂₉, C₃⋊C₈×C₂₉, …
derived subgroup of	#90773, D₂₃₂, Dic₁₁₆, …
Frattini subgroup of	Dic₁₀₀₉₂, D₂₀₁₈₄
Fitting subgroup of	#6597, Dic₅₈, C₄×D₂₉, …
maximal subgroup of	#6597, C₈×C₂₉, Dic₅₈, …
maximal normal subgroup of	#6597, C₈×C₂₉, Dic₅₈, …
maximal quotient of	C₈×C₂₉, C₂×C₄×C₂₉, Dic₃×C₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(116, 2);