Finite Group Databasebeta

← Prev (#72244457) Next (#72244459) →

AGL(1,𝔽5)×#11713200 #72244458

order15360
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productAGL(1,𝔽5)×#11713200

Splittings

FittingC5×#2846 ⋊ C4×D3
SylowC5 ⋊ C4×#11713200

Subgroups

subgroupG/H
centerC2C2×AGL(1,𝔽5)×#1020⋊C2
commutatorC2×C5×C22⋊A4C23×C4
FrattiniC23C22×AGL(1,𝔽5)×S4
FittingC5×#2846C4×D3
socleC23×C5C22×C4×S4
prime radicalD5×#11713200C2
cosocleC2×D5×C22⋊A4C24
hypercenterC22AGL(1,𝔽5)×#1020⋊C2
hypocenterC5×C22⋊A4C2×C4×D4
LCSG ▷ C2×C5×C22⋊A4 ▷ C5×C22⋊A4
UCSC1 ◁ C2 ◁ C22

At least 24172 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 4 generators on 17 points:

(2,3,4,5)(11,14,16)(12,13,17);(1,2,5,3,4)(7,9);(13,17)(14,15);(6,7)(8,9)(10,11)(12,16)(13,14)(15,17)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

automorphism group ofC2×Q8×D5, C2×#713

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,3,4,5)(11,14,16)(12,13,17), (1,2,5,3,4)(7,9), (13,17)(14,15), (6,7)(8,9)(10,11)(12,16)(13,14)(15,17));