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C52×(C43⋊C3)⋊C4 #27166373

order12900
automorphism?
homologyAbelianization (H1) = C3×C4×C52 · Schur multiplier (H2) = C5

Aliases

direct productC52×(C43⋊C3)⋊C4
unambiguous semidirect product(C43⋊C3)⋊(C4×C52)
unambiguous semidirect productDic43⋊(C3×C52)
unambiguous semidirect product(C5×Dic43)⋊(C3×C5)

Splittings

derived, SylowC43 ⋊ C3×C4×C52
HallC52×Dic43 ⋊ C3
HallC52×C43⋊C3 ⋊ C4
HallC52×C43 ⋊ C3×C4
HallDic43 ⋊ C3×C52
HallC43⋊C3 ⋊ C4×C52

Subgroups

subgroupG/H
center, hypercenterC2×C52D43⋊C3
commutator, hypocenterC43C3×C4×C52
FrattiniC2C52×D43⋊C3
Fitting, socleC2×C52×C43C2×C3
prime radicalC2×C52×C43⋊C3C2
cosocleC2×C43C2×C3×C52
LCSG ▷ C43
UCSC1 ◁ C2×C52

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 4300

A minimal faithful permutation representation — 2 generators on 57 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 42);