Finite Group Databasebeta

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Cayley diagram

C2×#5005 #82221

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#5005

Splittings

FittingC2×C3×#117 ⋊ C2
SylowC3 ⋊ C2×#421

Subgroups

subgroupG/H
centerC2×C4C2×D12
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC2×C3×#117C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#235C2
cosocleC2×C3×C4C24
hypercenterC2×#117D3
hypocenterC3C2×#421
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

772 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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10700648, #10700650, #10700655, …
maximal normal subgroup of#10700648, #10700650, #10700655, …
maximal quotient ofC2×#70757, C2×#70767, C2×#70755, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14797);