Finite Group Databasebeta

← Prev (#10631980) Next (#10631982) →

C3×#13374 #10631981

order768
nilpotency class4
isoclinismisoclinic class of #13374
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC3×#13374

Subgroups

subgroupG/H
center, socleC2×C3#2023
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#13374, #72271585, #72271639, …
maximal subgroup ofC3×#500940, D3×#13374, #72271585, …
maximal normal subgroup ofC3×#500940, D3×#13374, #72271585, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6362);