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C2×C5×D1221 #37567643

order24420
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×D1221
unambiguous semidirect product(C2×C3)⋊(C5×D407)
unambiguous semidirect product(C2×C37)⋊(C5×D33)

Splittings

Fitting, socleC2×C3×C5×C11×C37 ⋊ C2
derived, HallC3×C11×C37 ⋊ C22×C5
HallC3×C5×C11×C37 ⋊ C22
HallC5×C11×C37 ⋊ C2×D3
HallC3×C5×C37 ⋊ C2×D11
HallC11×C37 ⋊ C2×C5×D3
HallC5×C37 ⋊ C2×D33
HallC3×C5×C11 ⋊ C2×D37
HallC3×C37 ⋊ C2×C5×D11
HallC5×C11 ⋊ C2×D111
SylowC37 ⋊ C2×C5×D33
HallC3×C11 ⋊ C2×C5×D37
HallC3×C5 ⋊ C2×D407
SylowC11 ⋊ C2×C5×D111
SylowC3 ⋊ C2×C5×D407

Subgroups

subgroupG/H
center, hypercenterC2×C5D1221
commutator, cosocle, hypocenterC3×C11×C37C22×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C11×C37C2
LCSG ▷ C3×C11×C37
UCSC1 ◁ C2×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 12210

A minimal faithful permutation representation — 2 generators on 58 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(39,40)(41,42)(43,44)(45,46)(47,48)(50,51)(52,53,54,55,56);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(38,39,41,43,45,47,48,46,44,42,40)(49,50,51)(57,58)

Conjugacy classes & character table

6120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24420, 164);