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C3×#16500 #10635107

order768
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C4

Aliases

direct productC3×#16500

Subgroups

subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

2698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72254736, #72254747, #72270408, …
maximal subgroup ofC3×#4312024, #72254736, #72254747, …
maximal normal subgroup ofC3×#4312024, #72254736, #72254747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 9488);