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C2×C3×#3018 #10678989

order768
nilpotency class3
isoclinismisoclinic class of #453
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC2×C3×#3018

Subgroups

subgroupG/H
centerC2×C3×C4#120
commutator, Frattini, cosocleC2×C4C25×C3
socleC22×C3C2×#120
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G

4850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 21 points:

(1,2)(19,20,21);(3,4)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(3,4)(5,9)(6,10)(7,15)(8,11)(12,16)(13,17)(14,18);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#10826480, C2×D3×#3018
maximal subgroup ofC2×#10826480, C2×D3×#3018, C2×C3×#61110
maximal normal subgroup ofC2×#10826480, C2×D3×#3018, C2×C3×#61110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53370);