Finite Group Databasebeta

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Cayley diagram

Dic3×#117 #82222

order384
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42

Aliases

direct productDic3×#117

Splittings

SylowC3 ⋊ C4×#117

Subgroups

subgroupG/H
centerC22×C4C22×D3
commutatorC2×C3C22×C42
FrattiniC23C23×D3
FittingC2×C3×#117C2
socle, cosocleC23×C3C24
prime radicalC24×C3C23
hypercenterC2×#117D3
hypocenterC3C4×#117
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#117

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10688628, #10688632, Dic3×H(3,ℤ/4), …
maximal normal subgroup of#10688628, #10688632, Dic3×H(3,ℤ/4), …
maximal quotient ofDic3×#376, C4×Dic3×C22⋊C4, C4×Dic3×C4⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14798);