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Cayley diagram

C5×C23 #1178

order115
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C4×C11 (order 88) · Inn(G) = C1 · Out(G) = C2×C4×C11
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽23)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC115
abelianC5×C23

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 115

A minimal faithful permutation representation — 1 generator on 28 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)

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

center ofC5×D3×C23, C5×D5×C23, C5×A4×C23, …
derived subgroup ofD115, Dic115, C23⋊AGL(1,𝔽5), …
Frattini subgroup ofC2×C25×C529, D25×C529, C25×D529, …
Fitting subgroup ofD5×C23, C5×D23, D115, …
socle ofD5×C23, C5×D23, D115, …
maximal subgroup ofD5×C23, C5×D23, D115, …
maximal normal subgroup ofD5×C23, C5×D23, D115, …
maximal quotient ofC2×C5×C23, C3×C5×C23, C23×C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(115, 1);