Finite Group Databasebeta

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Cayley diagram

C3×#520 #5850

order192
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphismAut(G) order 6144
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC3×#520

Subgroups

subgroupG/H
center, socleC22×C3C24
commutatorC2C23×C3×C4
Frattini, cosocleC22C24×C3
prime radicalC2×C3×#41C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#71664, #71665, #71666, …
maximal subgroup of#71664, #71665, #71666, …
maximal normal subgroup of#71664, #71665, #71666, …
maximal quotient ofC3×#2274, C3×#2275, C3×#2276, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1409);