Finite Group Databasebeta

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Cayley diagram

C3×#381 #5255

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 8192
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productC3×#381

Subgroups

subgroupG/H
center, socleC23×C3C23
commutatorC22C22×C3×C4
Frattini, cosocleC23C23×C3
prime radicalC24×C3C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67627, #67628, #67629, …
maximal subgroup of#67627, #67628, #67629, …
maximal normal subgroup of#67627, #67628, #67629, …
maximal quotient ofC3×C22⋊#95, C3×#1434, C3×#1440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 814);