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H(3,𝔽3)×#126 #11721914

order864
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)×#120
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C32

Aliases

direct productH(3,𝔽3)×#126

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC22×C3C23×C32
prime radicalC23×H(3,𝔽3)C22
LCSG ▷ C22×C3 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

330 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)(11,15)(12,19)(17,24)(18,22)(20,23)(21,25);(2,5,6)(4,8,9)(10,11,16,22)(12,17,23,24)(13,15,14,18)(19,25,20,21);(10,12,13,20)(11,17,15,21)(14,19,16,23)(18,25,22,24)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#10596172⋊C3, #10596173⋊C3, #10596181⋊C3, …
maximal subgroup of#10596172⋊C3, #10596173⋊C3, #10596181⋊C3, …
maximal normal subgroup of#10596172⋊C3, #10596173⋊C3, #10596181⋊C3, …
maximal quotient ofH(3,𝔽3)×#383, H(3,𝔽3)×#387, H(3,𝔽3)×#388, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 1881);