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C3×#60141 #10678748

order768
nilpotency class2
isoclinismisoclinic class of #3484
automorphism?
homologyAbelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29

Aliases

direct productC3×#60141

Subgroups

subgroupG/H
centerC2×C3×C4C25
commutatorC22C24×C3×C4
Frattini, cosocleC2×C4C25×C3
socleC22×C3C24×C4
prime radicalC2×C3×#525C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

2862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#60141, #72249320, #72253933, …
maximal subgroup ofD3×#60141, C3×#364424, #72249320, …
maximal normal subgroup ofD3×#60141, C3×#364424, #72249320, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53129);