Finite Group Databasebeta

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Cayley diagram

C3×#380 #5254

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 4096
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productC3×#380

Subgroups

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

234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67624, #67625, #67626, …
maximal subgroup of#67624, #67625, #67626, …
maximal normal subgroup of#67624, #67625, #67626, …
maximal quotient ofC3×C22⋊#95, C3×#1434, C3×C4⋊#95, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 813);