Finite Group Databasebeta

← Prev (#10679773) Next (#10679775) →

C3×#61167 #10679774

order768
nilpotency class3
isoclinismisoclinic class of #61049
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#61167

Subgroups

subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#525 ◁ G

2286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 27 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#61167, #72249270
maximal subgroup ofC3×#10576928, D3×#61167, #72249270
maximal normal subgroup ofC3×#10576928, D3×#61167, #72249270

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 54155);