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Cayley diagram

C4×#1007 #85793

order384
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC4×#1007

Splittings

FittingC2×C3×C4×Q8 ⋊ C2
SylowC3 ⋊ C4×2−1+4

Subgroups

subgroupG/H
centerC2×C4C23×D3
commutatorC2×C3C24×C4
FrattiniC22C24×D3
FittingC2×C3×C4×Q8C2
socle, cosocleC22×C3C25
prime radicalC2×#1007C2
hypercenterC2×C4×Q8D3
hypocenterC3C4×2−1+4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10686590, #10686593, #10761571, …
maximal normal subgroup of#10686590, #10686593, #10761571, …
maximal quotient ofC4×#5504, C4×#5574, C4×#5572, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18369);