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Cayley diagram

C17×SL(2,𝔽3) #87977

order408
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C16×S4 (order 384) · Inn(G) = A4 · Out(G) = C2×C16
homologyAbelianization (H1) = C3×C17 · Schur multiplier (H2) = C1

Aliases

direct productC17×SL(2,𝔽3)
unambiguous semidirect productQ8⋊(C3×C17)
unambiguous semidirect product(Q8×C17)⋊C3

Splittings

Fitting, HallQ8×C17 ⋊ C3
derived, SylowQ8 ⋊ C3×C17

Subgroups

subgroupG/H
center, socle, hypercenterC2×C17A4
commutator, cosocle, hypocenterQ8C3×C17
FrattiniC2A4×C17
FittingQ8×C17C3
LCSG ▷ Q8
UCSC1 ◁ C2×C17

14 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 136

A minimal faithful permutation representation — 2 generators on 25 points:

(2,3,5)(6,8,7)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25);(1,5,4,7)(2,8,6,3)

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC17⋊#226, C17⋊GL(2,𝔽3), #13108921, …
maximal subgroup ofC17×#226, C17×GL(2,𝔽3), C17⋊#226, …
maximal normal subgroup ofC17×#226, C17×GL(2,𝔽3), C17⋊#226, …
maximal quotient ofC2×C17×SL(2,𝔽3), C17×Q8⋊C9, C3×C17×SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(408, 14);