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(C13⋊C3)⋊D413 #44567291

order32214
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product(C13⋊C3)⋊D413
unambiguous semidirect productC59⋊((C13⋊C3)⋊D7)

Splittings

Fitting, socle, derived, HallC7×C13×C59 ⋊ C2×C3
HallC7×C13⋊C3×C59 ⋊ C2
HallD5369 ⋊ C3
HallC13⋊C3×C59 ⋊ D7
HallC13×C59 ⋊ C3×D7
HallC7×C59 ⋊ D13⋊C3
HallC7×C13⋊C3 ⋊ D59
HallC7×C13 ⋊ C3×D59
SylowC59 ⋊ (C13⋊C3)⋊D7
HallC13⋊C3 ⋊ D413
SylowC13 ⋊ C3×D413
SylowC7 ⋊ (C13⋊C3)⋊D59

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C13×C59C2×C3
LCSG ▷ C7×C13×C59
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 79
transitive degreeμt(G) = 5369

A minimal faithful permutation representation — 2 generators on 79 points:

(2,3)(4,5)(6,7)(9,13,14,10,11,15)(12,19,18,16,17,20)(22,23)(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)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79);(1,2,4,6,7,5,3)(8,9,12,11,15,18,17,19,20,14,13,16,10)(21,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23)

Polycyclic presentation:

2,3,7,13,59|c^a=c^6,d^a=d^12,e^a=e^58,d^b=d^3

Conjugacy classes & character table

1450 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32214, 45);