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C4×C52⋊(C43⋊C3) #27166388

order12900
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C5

Aliases

direct productC4×C52⋊(C43⋊C3)
unambiguous semidirect product(C4×C52)⋊(C43⋊C3)
unambiguous semidirect product(C4×C43)⋊(C52⋊C3)

Splittings

Fitting, HallC4×C52×C43 ⋊ C3
derived, HallC52×C43 ⋊ C3×C4
HallC4×C43 ⋊ C52⋊C3
HallC4×C52 ⋊ C43⋊C3
SylowC43 ⋊ C4×C52⋊C3
SylowC52 ⋊ C4×C43⋊C3

Subgroups

subgroupG/H
center, hypercenterC4C52⋊(C43⋊C3)
commutator, hypocenterC52×C43C3×C4
FrattiniC2C2×C52⋊(C43⋊C3)
FittingC4×C52×C43C3
socle, cosocleC2×C52×C43C2×C3
prime radicalC2×C52⋊(C43⋊C3)C2
LCSG ▷ C52×C43
UCSC1 ◁ C4

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 62
transitive degreeμt(G) = 2580

A minimal faithful permutation representation — 2 generators on 62 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)(44,45,47)(46,50,52)(48,53,56)(49,55,58)(51,54,57)(59,60,61,62);(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)(44,46,51,56,58)(45,50,54,48,49)(47,53,52,55,57)

Conjugacy classes & character table

1444 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 57);