Finite Group Databasebeta

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Cayley diagram

Q8×C3⋊D3 #3831

order144
automorphismAut(G) = C2×S4×AGL(2,𝔽3) (order 20736) · Inn(G) = C22×C3⋊D3 · Out(G) = C2×D3×S4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22×C3

Aliases

direct productQ8×C3⋊D3

Splittings

FittingC32×Q8 ⋊ C2
SylowC32 ⋊ C2×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×C3⋊D3
commutator, socle, cosocleC2×C32C23
FittingC32×Q8C2
prime radicalC2×C3⋊D3C22
hypercenterQ8C3⋊D3
hypocenterC32C2×Q8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ Q8

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup ofD32⋊(#41⋊C3), #157⋊(#41⋊C3), SL(2,𝔽3)×GO+(4,𝔽2), …
maximal subgroup of#63784, #63940, #63944, …
maximal normal subgroup of#63784, #63940, #63944, …
maximal quotient of#64096, #64097, #64098, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 174);