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C2×AGL(1,𝔽5)×#1020⋊C2 #72243273

order7680
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×AGL(1,𝔽5)×#1020⋊C2

Splittings

FittingC2×C5×#120 ⋊ C4×D3
derivedC5×C22⋊A4 ⋊ C23×C4
SylowC5 ⋊ C2×C4×#1020⋊C2

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽5)×#1020⋊C2
commutator, hypocenterC5×C22⋊A4C23×C4
FrattiniC22C22×AGL(1,𝔽5)×S4
FittingC2×C5×#120C4×D3
socleC23×C5C2×C4×S4
prime radicalC2×D5×#1020⋊C2C2
cosocleD5×C22⋊A4C24
LCSG ▷ C5×C22⋊A4
UCSC1 ◁ C2

11408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#11713200
maximal normal subgroup ofAGL(1,𝔽5)×#11713200
maximal quotient ofAGL(1,𝔽5)×#11713200

GAP

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

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