Finite Group Databasebeta

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Cayley diagram

C2×#4506 #71405

order384
isoclinismisoclinic class of D48
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×#4506

Splittings

SylowC3 ⋊ C2×#368

Subgroups

subgroupG/H
centerC2×C4D24
commutatorC3×C8C22×C4
FrattiniC2×C8C22×D3
FittingC22×C3×C16C2
socleC22×C3#107
prime radicalC23×C3Q16
cosocleC2×C3×C8C23
hypercenterC22×C16D3
hypocenterC3C2×#368
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16

144 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)(6,8,18,34,11,21,25,35,12,22,23,27,9,19,10,20)(7,13,24,31,14,28,30,33,16,29,37,36,26,32,15,17);(4,5)(6,7,9,14,12,16,11,26)(8,17,19,31,22,33,21,36)(10,24,23,30,25,37,18,15)(13,27,28,35,29,34,32,20);(1,2)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10682889, #10682890, #10682891, …
maximal normal subgroup of#10682889, #10682890, #10682891, …
maximal quotient ofC2×#67500, C2×#67501, #10706772, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3981);