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Cayley diagram

C2×C3⋊#401 #73348

order384
isoclinismisoclinic class of #4656
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×C3⋊#401
unambiguous semidirect product(C2×C3)⋊#401

Splittings

SylowC3 ⋊ C2×#401

Subgroups

subgroupG/H
centerC24C22×D3
commutator, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC2×C3×#95C2
socle, prime radicalC24×C3C23
hypercenterC2×#95D3
hypocenterC3C2×#401
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#95

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10717049, C4×C3⋊#401, #10721498, …
maximal normal subgroup of#10717049, C4×C3⋊#401, #10721498, …
maximal quotient ofC2×#69418, C2×#69417, C2×#69416, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5924);