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Cayley diagram

C2×#985 #5921

order192
isoclinismisoclinic class of #226
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = S4 · Out(G) = #120
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×#985

Splittings

FittingC2×#41 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2×C4S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23
FrattiniC2C22×S4
FittingC2×#41D3
socleC22C2×S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4

153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73127, C4×#985, #85341, …
maximal normal subgroup of#73127, C4×#985, #85341, …
maximal quotient ofC2×C4×#226, C2×#5389, C2×C4×GL(2,𝔽3), …
outer automorphism group ofC3⋊Dic15, C2×C3⋊D15

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1480);