Finite Group Databasebeta

← Prev (#10652244) Next (#10652246) →

C3×#33638 #10652245

order768
nilpotency class4
isoclinismisoclinic class of #33638
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#33638

Subgroups

subgroupG/H
center, socleC2×C3C2×#476
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC2×C3×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 35 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#33638, #72271301, #72271602, …
maximal subgroup ofC3×#7623148, D3×#33638, #72271301, …
maximal normal subgroup ofC3×#7623148, D3×#33638, #72271301, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26626);