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C17×SL(2,ℤ/4) #11717830

order816
isoclinismisoclinic class of S4
automorphism?
homologyAbelianization (H1) = C4×C17 · Schur multiplier (H2) = C2

Aliases

direct productC17×SL(2,ℤ/4)
unambiguous semidirect productA4⋊(C4×C17)

Splittings

derivedA4 ⋊ C4×C17

Subgroups

subgroupG/H
center, hypercenterC2×C17S4
commutator, hypocenterA4C4×C17
FrattiniC2C17×S4
Fitting, socleC23×C17D3
prime radicalC2×A4×C17C2
cosocleC2×A4C2×C17
LCSG ▷ A4
UCSC1 ◁ C2×C17

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 204

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

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13109627, #13109629, D17×SL(2,ℤ/4), …
maximal normal subgroup of#13109627, #13109629, D17×SL(2,ℤ/4), …
maximal quotient ofC17×A4⋊C8, C17×#859, C17×GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(816, 100);