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Cayley diagram

D3×C4⋊C8 #4832

order192
automorphismAut(G) = C23×D3×#120 (order 1536) · Inn(G) = C22×D3 · Out(G) = C23×D4
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productD3×C4⋊C8

Splittings

FittingC3×C4⋊C8 ⋊ C2
SylowC3 ⋊ C2×C4⋊C8

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×D3
commutatorC2×C3C22×C8
FittingC3×C4⋊C8C2
socleC22×C3C22×C4
prime radicalC22×D3C2×C4
cosocleC2×C3×C4C23
hypercenterC4⋊C8D3
hypocenterC3C2×C4⋊C8
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4⋊C8

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D4⋊C8, #68166, #68168, …
maximal normal subgroup ofD3×D4⋊C8, #68166, #68168, …
maximal quotient ofD3×#334, #68527, #68528, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 391);