Finite Group Databasebeta

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Cayley diagram

C2×#5144 #83790

order384
isoclinismisoclinic class of #213
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#5144

Splittings

FittingC2×C3×#131 ⋊ C2
SylowC3 ⋊ C2×#443

Subgroups

subgroupG/H
centerC2×C8#67
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC2×C3×#131C2
socleC22×C3C4×D4
prime radicalC2×#950C2
cosocleC2×C3×C4C24
hypercenterC2×#131D3
hypocenterC3C2×#443
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#131

472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(3,4,5)(6,9,10,19,11,20,21,12)(7,8,13,16,15,18,14,17);(1,2)(6,8)(7,12)(9,13)(10,16)(11,18)(14,20)(15,19)(17,21);(4,5)(6,7)(8,9)(10,13)(11,15)(12,17)(14,21)(16,19)(18,20);(1,2)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10712061, #10712062, #10712064, …
maximal normal subgroup of#10712061, #10712062, #10712064, …
maximal quotient ofC2×C8×#213, C2×C8×#214, C2×C8×#215, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16366);