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C3×#63041 #10681648

order768
nilpotency class2
isoclinismisoclinic class of #63041
automorphism?
homologyAbelianization (H1) = C26×C3 · Schur multiplier (H2) = C213

Aliases

direct productC3×#63041

Subgroups

subgroupG/H
center, socleC22×C3C26
commutator, Frattini, cosocleC22C26×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

7428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 27 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#63041, #72247954
maximal subgroup ofC3×#511827, D3×#63041, #72247954
maximal normal subgroup ofC3×#511827, D3×#63041, #72247954
maximal quotient ofC3×#7918389, C3×#7837029

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 56029);