Finite Group Databasebeta

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Cayley diagram

C3×#408 #5282

order192
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) = C22×#23900 (order 1024) · Inn(G) = C23 · Out(G) = C24×D4
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×#408

Subgroups

subgroupG/H
centerC2×C3×C4C23
commutatorC22C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socleC22×C3C22×C4
prime radicalC2×C3×#41C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×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)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22,23);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67689, #67690, #67691, …
maximal subgroup of#67689, #67690, #67691, …
maximal normal subgroup of#67689, #67690, #67691, …
maximal quotient ofC3×#1468, C3×#1469, C3×#1470, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 841);