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C3×#7548 #10626155

order768
nilpotency class6
isoclinismisoclinic class of #7548
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×#7548

Subgroups

subgroupG/H
center, socleC2×C3#1424
commutatorC32C2×C3×C4
Frattini, cosocleC2×C32C22×C3
prime radicalC22×C3SD64
LCSG ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ C2×C3×C32 ◁ G

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 2 generators on 67 points:

(1,2,3)(5,11,13,29)(6,17)(7,22)(8,23)(9,26)(12,33)(14,42,39,56)(15,46,40,66)(16,47,41,32)(18,25)(19,36)(20,38)(21,37)(24,51)(27,54)(28,55,44,30)(31,59,58,43)(34,61,62,57)(35,64)(45,67,65,63)(48,49)(50,60)(52,53);(4,5,12,34,7,14,35,58,23,43,50,56,53,61,21,29,9,16,37,55,24,45,60,66,27,40,64,67,18,28,33,47,10,13,38,62,25,39,19,31,54,59,49,42,51,57,6,11,26,41,17,30,52,65,48,46,8,15,36,63,22,44,20,32)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#7548, #72255019
maximal subgroup ofC3×#153598, D3×#7548, #72255019
maximal normal subgroup ofC3×#153598, D3×#7548, #72255019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 536);