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C5×Dic5×C43⋊C3 #27166375

order12900
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×Dic5×C43⋊C3
unambiguous semidirect product(Dic5×C43)⋊(C3×C5)
unambiguous semidirect product(C5×Dic5×C43)⋊C3

Splittings

derivedC5×C43 ⋊ C3×C4×C5
HallC5×Dic5×C43 ⋊ C3
HallC52×C43⋊C3 ⋊ C4
HallC52×C43 ⋊ C3×C4
SylowC43 ⋊ C3×C5×Dic5
SylowC52 ⋊ C4×C43⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C5D5×C43⋊C3
commutator, hypocenterC5×C43C3×C4×C5
FrattiniC2C5×D5×C43⋊C3
Fitting, socleC2×C52×C43C2×C3
prime radicalC2×C52×C43⋊C3C2
cosocleC2×C5×C43C2×C3×C5
LCSG ▷ C5×C43
UCSC1 ◁ C2×C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 860

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 44);