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Cayley diagram

C7×D23 #66305

order322
isoclinismisoclinic class of D23
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×D23
unambiguous semidirect productC23⋊(C2×C7)

Splittings

Fitting, socle, HallC7×C23 ⋊ C2
derived, SylowC23 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC7D23
commutator, cosocle, hypocenterC23C2×C7
FrattiniC1G
Fitting, socleC7×C23C2
LCSG ▷ C23
UCSC1 ◁ C7

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 161

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30);(8,9,11,13,15,17,19,21,23,25,27,29,30,28,26,24,22,20,18,16,14,12,10)

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×D23, C2×C7×D23, ΓL(1,𝔽8)×D23, …
maximal normal subgroup ofD7×D23, C2×C7×D23, ΓL(1,𝔽8)×D23, …
maximal quotient ofC7×Dic23, C2×C7×D23, C3×C7×D23, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(322, 2);