Finite Group Databasebeta

← Prev (#1267) Next (#1269) →

Cayley diagram

C127 #1268

order127
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C7×C9 (order 126) · Inn(G) = C1 · Out(G) = C2×C7×C9
holomorphG ⋊ Aut(G) = AGL(1,𝔽127)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC127

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 127
transitive degreeμt(G) = 127

A minimal faithful permutation representation — 1 generator on 127 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127)

Conjugacy classes & character table

127 conjugacy classes — view the full character table.

Relationships

center ofD3×C127, D5×C127, A4×C127, …
derived subgroup ofD127, C127⋊C3, Dic127, …
Frattini subgroup ofD16129, C2×C16129, C2×D16129
Fitting subgroup ofD127, C127⋊C3, D127⋊C3, …
socle ofD127, C127⋊C3, D127⋊C3, …
maximal subgroup ofD127, C2×C127, C127⋊C3, …
maximal normal subgroup ofD127, C2×C127, C127⋊C3, …
maximal quotient ofC2×C127, C3×C127, C5×C127, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(127, 1);