Finite Group Databasebeta

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Cayley diagram

C23×#217 #87300

order384
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C211

Aliases

direct productC23×#217

Splittings

SylowC3 ⋊ C23×C22⋊C4

Subgroups

subgroupG/H
centerC25C2×D3
commutatorC2×C3C24×C4
FrattiniC22C24×D3
Fitting, prime radicalC26×C3C2
socleC25×C3C22
cosocleC22×C3C25
hypercenterC26D3
hypocenterC3C23×C22⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C25 ◁ C26

4632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10683109, #10700448, #10700484, …
maximal normal subgroup of#10683109, #10700448, #10700484, …
maximal quotient ofC23×C3⋊#95, C22×C4×#217, C22×#5247, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19876);