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C3×#60387 #10678994

order768
nilpotency class3
isoclinismisoclinic class of #3019
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC3×#60387

Subgroups

subgroupG/H
centerC3×C4C2×#120
commutator, Frattini, cosocleC2×C4C25×C3
socleC2×C3C22×#120
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G

4764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 35 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#60387, #72253891
maximal subgroup ofD3×#60387, #72253891, C3×#6437576
maximal normal subgroup ofD3×#60387, #72253891, C3×#6437576

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53375);