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C23⋊#13095538 #49088385

order35742
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC23⋊#13095538

Splittings

Fitting, socle, derived, HallC7×C23×C37 ⋊ C2×C3
HallC23×#11715912 ⋊ C2
HallD5957 ⋊ C3
HallC23×C37 ⋊ AGL(1,𝔽7)
Hall#11715912 ⋊ D23
HallC7×C37 ⋊ C3×D23
HallC7×C23 ⋊ D37⋊C3
SylowC37 ⋊ ΓL(1,𝔽8)⋊D23
SylowC23 ⋊ #13095538
SylowC7 ⋊ C23⋊(D37⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C23×C37C2×C3
LCSG ▷ C7×C23×C37
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 5957

A minimal faithful permutation representation — 2 generators on 67 points:

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

Polycyclic presentation:

2,3,7,23,37|c^a=c^6,d^a=d^22,e^a=e^36,c^b=c^2,e^b=e^10

Conjugacy classes & character table

1028 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35742, 48);