Finite Group Databasebeta

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Cayley diagram

C22×C3×SD16 #5900

order192
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 12288
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC22×C3×SD16

Subgroups

subgroupG/H
center, socleC23×C3D4
commutator, Frattini, cosocleC4C24×C3
prime radicalC22×C3×D4C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ G

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72047, #72048, #72049, …
maximal subgroup of#72047, #72048, #72049, …
maximal normal subgroup of#72047, #72048, #72049, …
maximal quotient ofC22×C3×#102, C22×C3×#103, C22×C3×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1459);