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C3×#17587 #10636194

order768
nilpotency class3
isoclinismisoclinic class of #17577
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC3×#17587

Subgroups

subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#525C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3×C4 ◁ G

1466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2,3)(4,5)(6,10,11,8)(7,9)(12,13)(14,22,24,27)(15,20)(16,21)(17,19)(18,23,26,25);(4,6)(5,8)(7,11)(9,10)(12,14)(13,18)(15,23)(16,25)(17,24)(19,26)(20,22)(21,27);(13,19)(18,26)(20,21)(22,27);(5,9)(8,10)(12,15,17,16)(13,20,19,21)(14,23,24,25)(18,22,26,27)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72271750, #72271755
maximal subgroup ofC3×#186814, #72271750, #72271755
maximal normal subgroup ofC3×#186814, #72271750, #72271755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 10575);