Finite Group Databasebeta

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Cayley diagram

C3×C4×C4⋊C4 #5252

order192
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 4096
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4

Aliases

direct productC3×C4×C4⋊C4
unambiguous semidirect productC4⋊(C3×C42)

Subgroups

subgroupG/H
centerC22×C3×C4C22
commutatorC2C2×C3×C42
Frattini, cosocleC23C23×C3
socle, prime radicalC23×C3C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

194 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)(13,14,15);(5,6,7,8)(10,12);(9,10,11,12)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67619, #67620, #67621, …
maximal subgroup of#67619, #67620, #67621, …
maximal normal subgroup of#67619, #67620, #67621, …
maximal quotient ofC3×C4×#95, C3×#1434, C3×C4⋊#95, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 811);