Finite Group Databasebeta

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Cayley diagram

C3×#360 #4607

order192
nilpotency class4
isoclinismisoclinic class of #360
automorphismAut(G) = C2×#3109 (order 256) · Inn(G) = #102 · Out(G) = C23
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#360

Subgroups

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

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

(2,6,7,15)(3,9)(4,12)(8,10,13,16)(11,14)(17,18,19);(1,2)(3,10)(4,13)(5,7)(6,14)(8,12)(9,15)(11,16)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#69246, #69247, #69248, …
maximal subgroup of#69246, #69247, #69248, …
maximal normal subgroup of#69246, #69247, #69248, …
maximal quotient ofC3×#1333, C3×#1334, C3×#1347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 166);