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C25×C22⋊(C43⋊C3) #27166364

order12900
automorphism?
homologyAbelianization (H1) = C3×C25 · Schur multiplier (H2) = C2

Aliases

direct productC25×C22⋊(C43⋊C3)
unambiguous semidirect productC22⋊(C25×C43⋊C3)
unambiguous semidirect productC43⋊(A4×C25)
unambiguous semidirect product(C22×C25)⋊(C43⋊C3)
unambiguous semidirect product(C25×C43)⋊A4

Splittings

Fitting, HallC22×C25×C43 ⋊ C3
derived, HallC22×C43 ⋊ C3×C25
HallC25×C43 ⋊ A4
HallC22×C25 ⋊ C43⋊C3
SylowC43 ⋊ A4×C25
SylowC22 ⋊ C25×C43⋊C3

Subgroups

subgroupG/H
center, hypercenterC25C22⋊(C43⋊C3)
commutator, hypocenterC22×C43C3×C25
FrattiniC5C5×C22⋊(C43⋊C3)
FittingC22×C25×C43C3
socle, cosocleC22×C5×C43C3×C5
prime radicalC5×C22⋊(C43⋊C3)C5
LCSG ▷ C22×C43
UCSC1 ◁ C25

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 4300

A minimal faithful permutation representation — 2 generators on 72 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)(45,47,46)(48,49,51,54,58,50,52,55,59,63,53,56,60,64,67,57,61,65,68,70,62,66,69,71,72);(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)(44,47)(45,46)

Conjugacy classes & character table

1500 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 33);