Finite Group Databasebeta

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Cayley diagram

C3×#362 #4609

order192
nilpotency class4
isoclinismisoclinic class of #360
automorphismAut(G) = C2×#61451 (order 512) · Inn(G) = #102 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#362

Subgroups

subgroupG/H
center, socleC2×C3#102
commutatorC8C2×C3×C4
Frattini, cosocleC2×C8C22×C3
prime radicalC22×C3SD16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2,5,9,6,11,21,10)(3,12,14,25,15,29,20,28)(4,16,18,8,19,26,32,27)(7,17,22,13,24,30,23,31)(33,34,35);(1,3,6,15)(2,7,11,24)(4,17,19,30)(5,20,21,14)(8,25,27,28)(9,23,10,22)(12,16,29,26)(13,18,31,32)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#69264, D3×#362, #69266, …
maximal subgroup of#69264, D3×#362, #69266, …
maximal normal subgroup of#69264, D3×#362, #69266, …
maximal quotient ofC3×#1334, C3×#1350, C3×#1362, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 168);