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Cayley diagram

C23⋊C11 #7008

order253
isoclinismisoclinic class of C23⋊C11
automorphismAut(G) = AGL(1,𝔽23) (order 506) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C11 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC23⋊C11

Splittings

Fitting, socle, derived, SylowC23 ⋊ C11

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC23C11
LCSG ▷ C23
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 23

A minimal faithful permutation representation — 2 generators on 23 points:

(2,3,4,6,9,10,15,18,5,8,12)(7,11,16,14,17,22,21,20,23,13,19);(1,2,3,5,4,7,8,13,6,10,11,17,12,18,19,23,9,14,15,20,16,21,22)

Polycyclic presentation:

11,23|b^a=b^2

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽23), C2×C23⋊C11, C3×C23⋊C11, …
maximal normal subgroup ofAGL(1,𝔽23), C2×C23⋊C11, C3×C23⋊C11, …
maximal quotient ofC2×C23⋊C11, C3×C23⋊C11, C5×C23⋊C11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(253, 1);