Finite Group Databasebeta

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Cayley diagram

C4×#919 #79717

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

Aliases

direct productC4×#919

Splittings

FittingC3×C4×Q16 ⋊ C2
SylowC3 ⋊ C4×#135

Subgroups

subgroupG/H
centerC2×C4D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×C4×Q16C2
socleC22×C3C22×D4
prime radicalC2×#919C2
cosocleC2×C3×C4C24
hypercenterC4×Q16D3
hypocenterC3C4×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q16

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(8,10,12,16)(9,14,13,22)(11,20,21,18)(15,23,19,17);(8,9,12,13)(10,17,16,23)(11,19,21,15)(14,20,22,18);(2,3)(9,13)(10,16)(15,19)(18,20);(4,5,6,7)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10704926, #10704931, #10704952, …
maximal normal subgroup of#10704926, #10704931, #10704952, …
maximal quotient ofC8×#919, #10764717, #10764936, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12293);