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C17⋊#14519613 #45684161

order33150
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC17⋊#14519613

Splittings

Fitting, socle, derived, HallC52×C13×C17 ⋊ C2×C3
HallC13×C17×C52⋊C3 ⋊ C2
HallC5⋊D1105 ⋊ C3
HallC17×C52⋊C3 ⋊ D13
HallC13×C52⋊C3 ⋊ D17
HallC52×C17 ⋊ C3×D13
HallC52×C13 ⋊ C3×D17
HallC13×C17 ⋊ (C5⋊D5)⋊C3
HallC52⋊C3 ⋊ D221
SylowC52 ⋊ C3×D221
SylowC17 ⋊ #14519613
SylowC13 ⋊ C17⋊((C5⋊D5)⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C13×C17C2×C3
LCSG ▷ C52×C13×C17
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 3315

A minimal faithful permutation representation — 2 generators on 45 points:

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

Polycyclic presentation:

2,3,5^2,13,17|c^a=c^4,d^a=d^4,e^a=e^12,f^a=f^16,c^b=cd,d^b=c^2d^3

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33150, 43);