Finite Group Databasebeta

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Cayley diagram

C3×#416 #5290

order192
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×#7630939 (order 1024) · Inn(G) = D4 · Out(G) = C22×#120
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×#416

Subgroups

subgroupG/H
centerC2×C3×C4D4
commutatorC4C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socleC22×C3C2×D4
prime radicalC2×C3×#41C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ G

158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4)(5,6)(7,14)(8,11)(9,13)(10,16)(12,18)(15,19)(17,20)(21,22,23);(5,7)(6,10)(8,15)(9,17)(11,14)(12,20)(13,19)(16,18);(5,8,9,18)(6,11,13,12)(7,15,17,16)(10,14,19,20)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#70287, #70288, #70289, …
maximal subgroup of#70287, #70288, #70289, …
maximal normal subgroup of#70287, #70288, #70289, …
maximal quotient ofC3×#1476, C3×#1482, C3×#1488, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 849);