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C3×#33692 #10652299

order768
nilpotency class4
isoclinismisoclinic class of #33691
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#33692

Subgroups

subgroupG/H
center, socleC2×C3#3109
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#405 ◁ G

536 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)(6,14)(7,18)(8,22)(10,23)(12,27)(13,26)(15,20)(16,31)(17,32)(19,33)(21,34)(24,35)(25,30)(28,29);(4,5,9,11)(6,10,16,25)(7,19,21,12)(8,13,22,26)(14,30,31,23)(15,29,17,24)(18,33,34,27)(20,35,32,28);(4,6)(5,10)(7,20)(8,15)(9,16)(11,25)(12,28)(13,29)(14,18)(17,22)(19,35)(21,32)(23,27)(24,26)(30,33)(31,34);(5,11)(6,15,16,17)(7,21)(10,24,25,29)(13,26)(14,20,31,32)(18,34)(23,35,30,28)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#33692, #72270600
maximal subgroup ofD3×#33692, #72270600, C3×#7623148
maximal normal subgroup ofD3×#33692, #72270600, C3×#7623148

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26680);