Finite Group Databasebeta

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Cayley diagram

C3×#383 #5257

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 12288
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×#383

Subgroups

subgroupG/H
center, socle, prime radicalC23×C3C23
commutatorC22C22×C3×C4
Frattini, cosocleC23C23×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67633, #69564, #69565, …
maximal subgroup of#67633, C3×#1288, #69564, …
maximal normal subgroup of#67633, C3×#1288, #69564, …
maximal quotient ofC3×#1434, C3×#1443, C3×#1839, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 816);