Finite Group Databasebeta

← Prev (#1024) Next (#1026) →

Cayley diagram

C97 #1025

order97
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C3×C32 (order 96) · Inn(G) = C1 · Out(G) = C3×C32
holomorphG ⋊ Aut(G) = AGL(1,𝔽97)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC97

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 97
transitive degreeμt(G) = 97

A minimal faithful permutation representation — 1 generator on 97 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)

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

center ofD3×C97, D5×C97, A4×C97, …
derived subgroup ofD97, C97⋊C3, Dic97, …
Frattini subgroup ofD9409, C2×C9409, C9409⋊C3, …
Fitting subgroup ofD97, C97⋊C3, #87604, …
socle ofD97, C97⋊C3, #87604, …
maximal subgroup ofD97, C2×C97, C97⋊C3, …
maximal normal subgroup ofD97, C2×C97, C97⋊C3, …
maximal quotient ofC2×C97, C3×C97, C5×C97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(97, 1);