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Cayley diagram

C2×GO+(4,𝔽2) #3843

order144
isoclinismisoclinic class of GO+(4,𝔽2)
automorphismAut(G) = #10596417 (order 576) · Inn(G) = GO+(4,𝔽2) · Out(G) = D4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×GO+(4,𝔽2)
unambiguous semidirect product#157⋊C22
unambiguous semidirect productD32⋊C22

Splittings

Fitting, socleC2×C32 ⋊ D4
SylowC32 ⋊ C2×D4

Subgroups

subgroupG/H
center, hypercenterC2GO+(4,𝔽2)
commutator, cosocleC3⋊D3C23
FrattiniC1G
Fitting, socleC2×C32D4
hypocenterC32C2×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 3 generators on 8 points:

(3,6,5,8,7,4);(6,8);(1,2)

Conjugacy classes & character table

18 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofGO+(4,𝔽2)⋊C4, C4×GO+(4,𝔽2), #64230, …
maximal normal subgroup ofGO+(4,𝔽2)⋊C4, C4×GO+(4,𝔽2), #64230, …
maximal quotient ofD32⋊Q8, #64221, #64222, …
automorphism group ofC3×D32
outer automorphism group ofC3×H(5,𝔽2), D3×H(5,𝔽2), #5966

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 186);