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C43⋊#64418 #27162445

order12642
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC43⋊#64418

Splittings

Fitting, socle, HallC72×C43 ⋊ D3
derived, HallC43×#3862 ⋊ C2
Hall#3862 ⋊ D43
SylowC72 ⋊ D129
SylowC43 ⋊ #64418

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC43×#3862C2
Fitting, socleC72×C43D3
LCSG ▷ C43×#3862
UCSC1

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 602

A minimal faithful permutation representation — 2 generators on 57 points:

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

Polycyclic presentation:

2,3,43,7^2|b^a=b^2,c^a=c^42,d^a=d^2e^3,e^a=d^6e^5,d^b=d^4e^4,e^b=e^2

Conjugacy classes & character table

419 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD43×#64418, #64418⋊D43, #38760426, …
maximal normal subgroup ofD43×#64418, #64418⋊D43, #38760426, …
maximal quotient ofC43⋊#10596868, C2×C43⋊#64418, C43⋊#11725071, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12642, 110);