Finite Group Databasebeta

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Cayley diagram

C23×D23 #67299

order368
isoclinismisoclinic class of D23
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC23×D23
unambiguous semidirect productC23⋊C24
unambiguous semidirect product(C2×C23)⋊C23

Splittings

Fitting, socleC23×C23 ⋊ C2
derived, SylowC23 ⋊ C24

Subgroups

subgroupG/H
center, hypercenterC23D23
commutator, cosocle, hypocenterC23C24
FrattiniC1G
Fitting, socleC23×C23C2
LCSG ▷ C23
UCSC1 ◁ C23

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 184

A minimal faithful permutation representation — 4 generators on 29 points:

(1,2)(3,4)(5,6)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29);(5,6)(7,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9);(3,4);(1,2)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22⋊C4×D23, #10625053, C2×(C2×D23)⋊C4, …
maximal normal subgroup ofC22⋊C4×D23, #10625053, C2×(C2×D23)⋊C4, …
maximal quotient ofC22×Dic46, C22×C4×D23, C22×D92, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 41);