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Cayley diagram

C2×2−1+4⋊C3 #5944

order192
automorphismAut(G) = S4×B(4,𝔽2) (order 1536) · Inn(G) = C22×A4 · Out(G) = C22×D4
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×2−1+4⋊C3
unambiguous semidirect product(C2×2−1+4)⋊C3

Splittings

Fitting, SylowC2×2−1+4 ⋊ C3

Subgroups

subgroupG/H
center, socleC22C22×A4
commutator, cosocle, hypocenterQ8C23×C3
FrattiniC2C23×A4
FittingC2×2−1+4C3
hypercenterC2×D4A4
LCSG ▷ Q8
UCSC1 ◁ C22 ◁ C2×D4

204 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)(3,4)(5,10,11,9,6,17)(7,8)(12,16,13,15,14,18);(3,11,7,13)(4,18,8,17)(5,14,12,6)(9,10,15,16);(1,2)

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73154, #73155, #73156, …
maximal normal subgroup of#73154, #73155, #73156, …
maximal quotient ofC2×#519⋊C3, C2×#520⋊C3, C4×2−1+4⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1503);