Finite Group Databasebeta

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Cayley diagram

C3×#504 #5378

order192
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = C22 · Out(G) = C22×C4×D3
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22

Aliases

direct productC3×#504

Subgroups

subgroupG/H
centerC3×C16C22
commutatorC2C22×C3×C8
Frattini, cosocleC8C23×C3
socleC2×C3C22×C8
prime radicalC3×#41C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C3×C16 ◁ G

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67823, #71451, #71452, …
maximal subgroup of#67823, C3×#1401, #71451, …
maximal normal subgroup of#67823, C3×#1401, #71451, …
maximal quotient ofC3×#2108, C3×#2113, C3×#2151, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 937);