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Cayley diagram

C3⋊D69 #88031

order414
isoclinismisoclinic class of C3⋊D69
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊D69
unambiguous semidirect productC23⋊(C3⋊D3)

Splittings

Fitting, socle, derived, HallC32×C23 ⋊ C2
SylowC23 ⋊ C3⋊D3
SylowC32 ⋊ D23

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C23C2
LCSG ▷ C32×C23
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 29 points:

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

Polycyclic presentation:

2,3^2,23|b^a=b^2,c^a=c^2,d^a=d^22

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊D3×D23, D3×D69, C2×C3⋊D69, …
maximal normal subgroup ofC3⋊D3×D23, D3×D69, C2×C3⋊D69, …
maximal quotient ofC3⋊Dic69, C2×C3⋊D69, C3⋊D207, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(414, 9);