Finite Group Databasebeta

← Prev (#11721928) Next (#11721930) →

H(3,𝔽3)×#131 #11721929

order864
nilpotency class2
isoclinismisoclinic class of D4×H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C32

Aliases

direct productH(3,𝔽3)×#131

Subgroups

subgroupG/H
centerC3×C8C22×C32
commutator, socleC2×C3C22×C32×C4
Frattini, cosocleC3×C4C23×C32
prime radical#41×H(3,𝔽3)C2
LCSG ▷ C2×C3 ▷ C1
UCSC1 ◁ C3×C8 ◁ G

341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#10589427⋊C3, #13135951, #13143256, …
maximal subgroup of#10589427⋊C3, #13135951, H(3,𝔽3)×#350, …
maximal normal subgroup of#10589427⋊C3, #13135951, H(3,𝔽3)×#350, …
maximal quotient ofH(3,𝔽3)×#405, H(3,𝔽3)×#408, H(3,𝔽3)×#424, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 1896);