Finite Group Databasebeta

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Cayley diagram

C2×C4⋊(C3⋊C8) #4923

order192
isoclinismisoclinic class of Dic6
automorphismAut(G) = C2×D3×#347042 (order 6144) · Inn(G) = C2×D3 · Out(G) = C2×#32898
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4⋊(C3⋊C8)

Splittings

SylowC3 ⋊ C2×C4⋊C8

Subgroups

subgroupG/H
centerC22×C4C2×D3
commutatorC2×C3C22×C8
FrattiniC2×C4C22×D3
FittingC2×C3×C42C2
socle, prime radicalC23×C3C2×C4
cosocleC2×C3×C4C23
hypercenterC2×C42D3
hypocenterC3C2×C4⋊C8
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#67619, #67634, #67654, …
maximal normal subgroup of#67619, #67634, #67654, …
maximal quotient ofC4×C4⋊(C3⋊C8), #69392, #69394, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 482);