Finite Group Databasebeta

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Cayley diagram

C3×#436 #5310

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) = C24×#120 (order 512) · Inn(G) = C23 · Out(G) = C26
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#436

Subgroups

subgroupG/H
centerC2×C3×C4C23
commutatorC22C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socleC22×C3C22×C4
prime radicalC2×C3×D4C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

122 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)(5,6,9,11,8,10,12,7)(13,14,15);(2,4)(6,10)(7,11);(5,7,9,6,8,11,12,10)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67761, #70859, #70860, …
maximal subgroup of#67761, C3×#1330, #70859, …
maximal normal subgroup of#67761, C3×#1330, #70859, …
maximal quotient ofC3×#1589, C3×#1590, C3×#1591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 869);