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#1044×C43⋊C3 #27166407

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

Aliases

direct product#1044×C43⋊C3
unambiguous semidirect product(C43×#1044)⋊C3

Splittings

Fitting, socle, derived, HallC52×C43 ⋊ C3×C4
HallC43×#1044 ⋊ C3
HallC52×C43⋊C3 ⋊ C4
SylowC43 ⋊ C3×#1044
SylowC52 ⋊ C4×C43⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC52×C43C3×C4
prime radicalC5⋊D5×C43⋊C3C2
cosocleC43×C5⋊D5C2×C3
LCSG ▷ C52×C43
UCSC1

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 430

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 76);