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Cayley diagram

C32×D23 #88027

order414
isoclinismisoclinic class of D23
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC32×D23
unambiguous semidirect productC23⋊(C2×C32)

Splittings

Fitting, socle, HallC32×C23 ⋊ C2
derived, SylowC23 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC32D23
commutator, cosocle, hypocenterC23C2×C32
FrattiniC1G
Fitting, socleC32×C23C2
LCSG ▷ C23
UCSC1 ◁ C32

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 207

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×D23, C3⋊D3×D23, C2×C32×D23, …
maximal normal subgroup ofC3×D3×D23, C3⋊D3×D23, C2×C32×D23, …
maximal quotient ofC32×Dic23, C2×C32×D23, C3×C9×D23, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(414, 5);