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Cayley diagram

C2×#4449 #69368

order384
isoclinismisoclinic class of D96
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×#4449

Splittings

FittingC2×C3×C32 ⋊ C2
SylowC3 ⋊ C2×SD64

Subgroups

subgroupG/H
centerC22D48
commutator, cosocleC3×C16C23
FrattiniC16C22×D3
FittingC2×C3×C32C2
socleC22×C3D16
prime radicalC2×D48C2
hypercenterC2×C32D3
hypocenterC3C2×SD64
LCSG ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10715765, C4×#4449, #10715777, …
maximal normal subgroup of#10715765, C4×#4449, #10715777, …
maximal quotient ofC2×#67596, C2×#67597, C2×#67592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1944);