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Cayley diagram

Dic3×Q32 #71495

order384
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productDic3×Q32

Splittings

SylowC3 ⋊ C4×Q32

Subgroups

subgroupG/H
centerC22D3×D8
commutatorC3×C8C22×C4
FrattiniC2×C8C22×D3
FittingC2×C3×Q32C2
socle, prime radicalC22×C3C2×D8
cosocleC2×C3×C8C23
hypercenterC2×Q32D3
hypocenterC3C4×Q32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q32

150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 3 generators on 39 points:

(5,7,10,18)(6,11,14,28)(8,20,23,36)(9,25,21,17)(12,30,31,35)(13,32,24,27)(15,33,34,19)(16,29,26,22)(37,38,39);(5,6,10,14)(7,15,18,34)(8,19,23,33)(9,24,21,13)(11,26,28,16)(12,29,31,22)(17,35,25,30)(20,32,36,27);(1,2,3,4)(38,39)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10715763, Dic3×SD64, #10715770, …
maximal normal subgroup of#10715763, Dic3×SD64, #10715770, …
maximal quotient ofC3⋊C8×Q32, #10707104, #10707178, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4071);