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Cayley diagram

C3×#349 #4596

order192
nilpotency class3
isoclinismisoclinic class of #99
automorphismAut(G) = C2×#3025 (order 256) · Inn(G) = C22⋊C4 · Out(G) = C24
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC3×#349

Subgroups

subgroupG/H
centerC3×C4C22⋊C4
commutatorC22C2×C3×C8
Frattini, cosocleC2×C8C22×C3
socleC2×C3C22⋊C8
prime radicalC23×C3C8
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67530, #67531, #67532, …
maximal subgroup of#67530, #67531, #67532, …
maximal normal subgroup of#67530, #67531, #67532, …
maximal quotient ofC3×#1314, C3×#1322, C3×#1330, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 155);