Finite Group Databasebeta

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Cayley diagram

C3×#359 #4606

order192
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C22×#442 (order 256) · Inn(G) = D8 · Out(G) = C22×C4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#359

Subgroups

subgroupG/H
centerC3×C4D8
commutatorC8C2×C3×C4
Frattini, cosocleC2×C8C22×C3
socleC2×C3#102
prime radicalC3×#135C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

56 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#69238, #69239, D3×#359, …
maximal subgroup of#69238, #69239, D3×#359, …
maximal normal subgroup of#69238, #69239, D3×#359, …
maximal quotient ofC3×#1331, C3×#1332, C3×#1349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 165);