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#1035×C43⋊C3 #27166346

order12900
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct product#1035×C43⋊C3
unambiguous semidirect product(C43×#1035)⋊C3

Splittings

Fitting, derived, HallC25×C43 ⋊ C3×C4
HallC43×#1035 ⋊ C3
HallC25×C43⋊C3 ⋊ C4
SylowC43 ⋊ C3×#1035
SylowC25 ⋊ C4×C43⋊C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, hypocenterC25×C43C3×C4
FrattiniC5AGL(1,𝔽5)×C43⋊C3
socleC5×C43C3×AGL(1,𝔽5)
prime radicalD25×C43⋊C3C2
cosocleC43×D25C2×C3
LCSG ▷ C25×C43
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 1075

A minimal faithful permutation representation — 2 generators on 68 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,33)(18,26,31)(21,29,28)(23,32,25)(27,36,40)(30,37,39)(34,41,35)(38,42,43)(45,47,48,53)(46,51,52,63)(49,57,58,62)(50,60,61,54)(55,65,66,68)(56,59,64,67);(1,2,4,8,15,20,5,9,16,25,35,32,12,19,28,24,34,41,22,31,39,36,14,23,33,40,43,42,26,7,13,21,30,38,37,17,3,6,11,18,27,29,10)(44,45,49,59,56,46,50,47,54,65,51,62,55,66,57,63,68,60,53,61,52,64,67,58,48)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 15);