Finite Group Databasebeta

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Cayley diagram

C3×#384 #5258

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 24576
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4

Aliases

direct productC3×#384

Subgroups

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

186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 192

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

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67634, #67635, #69567, …
maximal subgroup of#67634, #67635, C3×#1296, …
maximal normal subgroup of#67634, #67635, C3×#1296, …
maximal quotient ofC3×C4⋊#95, C3×#1444, C3×#1842, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 817);