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C5×(C43⋊C3)⋊Dic5 #27166374

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

Aliases

direct productC5×(C43⋊C3)⋊Dic5
unambiguous semidirect product(C43⋊C3)⋊(C5×Dic5)
unambiguous semidirect productDic215⋊(C3×C5)
unambiguous semidirect product(C5×Dic215)⋊C3

Splittings

derivedC5×C43 ⋊ C3×C4×C5
HallC5×Dic215 ⋊ C3
HallC52×C43⋊C3 ⋊ C4
HallC52×C43 ⋊ C3×C4
HallC43⋊C3 ⋊ C5×Dic5
SylowC43 ⋊ C3×C5×Dic5
SylowC52 ⋊ (C43⋊C3)⋊C4

Subgroups

subgroupG/H
center, hypercenterC2×C5C5⋊(D43⋊C3)
commutator, hypocenterC5×C43C3×C4×C5
FrattiniC2C5×C5⋊(D43⋊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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 43);