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C3×#33550 #10652157

order768
nilpotency class4
isoclinismisoclinic class of #2199
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#33550

Subgroups

subgroupG/H
centerC3×C4B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C3
socleC2×C3C2×B(4,𝔽2)
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G

1144 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)(4,5)(6,14)(7,11)(8,12)(9,13)(10,17)(15,26)(16,27)(18,23)(19,24)(20,25)(21,29)(22,30)(28,34)(31,33)(32,35);(4,6)(5,10)(7,16)(8,17)(9,15)(11,22)(12,14)(13,21)(18,30)(19,28)(20,29)(23,27)(24,32)(25,26)(31,35)(33,34);(6,15)(8,20)(10,21)(14,26)(16,28)(18,31)(22,32)(27,34);(4,7,9,19)(5,11,13,24)(6,16,15,28)(8,18,20,31)(10,22,21,32)(12,23,25,33)(14,27,26,34)(17,30,29,35)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#33550, #72247871, #72253895
maximal subgroup ofD3×#33550, #72247871, C3×#511827, …
maximal normal subgroup ofD3×#33550, #72247871, C3×#511827, …
maximal quotient ofC3×#498885

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26538);