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Cayley diagram

C2×#4521 #71492

order384
isoclinismisoclinic class of #4519
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×#4521

Splittings

FittingC2×C3×Q32 ⋊ C2
SylowC3 ⋊ C2×SD64

Subgroups

subgroupG/H
centerC22#826
commutator, cosocleC3×C16C23
FrattiniC16C22×D3
FittingC2×C3×Q32C2
socleC22×C3D16
prime radicalC2×D48C2
hypercenterC2×Q32D3
hypocenterC3C2×SD64
LCSG ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×Q32

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

(3,4,5)(6,7,11,15)(8,17,20,34)(9,21,23,37)(10,24,22,14)(12,25,27,16)(13,29,30,18)(19,36,32,28)(26,33,31,35);(4,5)(7,12)(8,16)(9,18)(10,22)(13,28)(14,31)(15,27)(17,21)(19,35)(20,25)(23,29)(24,26)(30,36)(32,33)(34,37);(1,2)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10715763, C4×#4521, #10715776, …
maximal normal subgroup of#10715763, C4×#4521, #10715776, …
maximal quotient ofC2×#67577, C2×#67826, C2×#67576, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4068);