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Cayley diagram

C3×Dic29 #66823

order348
isoclinismisoclinic class of D29
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×Dic29

Splittings

derived, SylowC29 ⋊ C3×C4
HallC3×C29 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C3D29
commutator, hypocenterC29C3×C4
FrattiniC2C3×D29
Fitting, socle, prime radicalC2×C3×C29C2
cosocleC2×C29C2×C3
LCSG ▷ C29
UCSC1 ◁ C2×C3

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 348

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(4,6,5,7)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#6599, C3⋊#6599, D3×Dic29, …
maximal normal subgroup ofC3×#6599, C3⋊#6599, D3×Dic29, …
maximal quotient ofC3×#6597, C2×C3×Dic29, C9×Dic29, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(348, 2);