Finite Group Databasebeta

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Cayley diagram

C2×#4824 #78978

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#4824

Splittings

FittingC2×C3×#104 ⋊ C2
SylowC3 ⋊ C2×#443

Subgroups

subgroupG/H
centerC2×C4D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC2×C3×#104C2
socleC22×C3C4×D4
prime radicalC2×#950C2
cosocleC2×C3×C4C24
hypercenterC2×#104D3
hypocenterC3C2×#443
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#4824, #10793738, #10793745, …
maximal normal subgroup ofC4×#4824, #10793738, #10793745, …
maximal quotient ofC2×#68148, C2×#68212, C2×#68149, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11554);