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C5×C3⋊D1023 #44501885

order30690
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×C3⋊D1023
unambiguous semidirect productC3⋊(C5×D1023)

Splittings

Fitting, socle, HallC32×C5×C11×C31 ⋊ C2
derived, HallC32×C11×C31 ⋊ C2×C5
HallC5×C11×C31 ⋊ C3⋊D3
HallC32×C5×C31 ⋊ D11
HallC32×C5×C11 ⋊ D31
HallC11×C31 ⋊ C5×C3⋊D3
HallC32×C31 ⋊ C5×D11
HallC5×C31 ⋊ C3⋊D33
HallC32×C11 ⋊ C5×D31
HallC5×C11 ⋊ C3⋊D93
HallC32×C5 ⋊ D341
SylowC31 ⋊ C5×C3⋊D33
SylowC11 ⋊ C5×C3⋊D93
SylowC32 ⋊ C5×D341

Subgroups

subgroupG/H
center, hypercenterC5C3⋊D1023
commutator, cosocle, hypocenterC32×C11×C31C2×C5
FrattiniC1G
Fitting, socleC32×C5×C11×C31C2
LCSG ▷ C32×C11×C31
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 15345

A minimal faithful permutation representation — 3 generators on 53 points:

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

Conjugacy classes & character table

7680 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30690, 252);