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Cayley diagram

C113 #1171

order113
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C7×C16 (order 112) · Inn(G) = C1 · Out(G) = C7×C16
holomorphG ⋊ Aut(G) = AGL(1,𝔽113)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC113

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C1

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 113
transitive degreeμt(G) = 113

A minimal faithful permutation representation — 1 generator on 113 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113)

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

center ofD3×C113, D5×C113, A4×C113, …
derived subgroup ofD113, Dic113, #90673, …
Frattini subgroup ofD12769, C2×C12769, C3×C12769, …
Fitting subgroup ofD113, #90673, C113⋊C7, …
socle ofD113, #90673, C113⋊C7, …
maximal subgroup ofD113, C2×C113, C3×C113, …
maximal normal subgroup ofD113, C2×C113, C3×C113, …
maximal quotient ofC2×C113, C3×C113, C5×C113, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(113, 1);