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Cayley diagram

C4×#916 #79644

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

Aliases

direct productC4×#916

Splittings

FittingC3×C4×SD16 ⋊ C2
SylowC3 ⋊ C4×#135

Subgroups

subgroupG/H
centerC2×C4D3×D4
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×C4×SD16C2
socleC22×C3C22×D4
prime radicalC2×#916C2
cosocleC2×C3×C4C24
hypercenterC4×SD16D3
hypocenterC3C4×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#916, #10764588, #10764665, …
maximal normal subgroup ofC8×#916, #10764588, #10764665, …
maximal quotient ofC8×#916, #10764713, #10764715, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12220);