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C53⋊#64424 #29510040

order15582
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC53⋊#64424

Splittings

Fitting, socle, derived, HallC72×C53 ⋊ C2×C3
HallC53×#3861 ⋊ C2
HallC7⋊D371 ⋊ C3
Hall#3861 ⋊ D53
SylowC53 ⋊ #64424
SylowC72 ⋊ C3×D53

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C53C2×C3
LCSG ▷ C72×C53
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 2597

A minimal faithful permutation representation — 3 generators on 67 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)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(55,58,59,56,57,60)(62,65,66,63,64,67);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(61,62,64,67,66,65,63);(54,55,57,60,59,58,56)

Polycyclic presentation:

2,3,7^2,53|c^a=c^6,d^a=d^6,e^a=e^52,c^b=c^2,d^b=d^2

Conjugacy classes & character table

508 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#23780749⋊C3, D53×#64424, #64423⋊D53, …
maximal normal subgroup of#23780749⋊C3, D53×#64424, #64423⋊D53, …
maximal quotient of#44528354, C2×C53⋊#64424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15582, 18);