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C23⋊((C37⋊C3)⋊D7) #49088382

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

Aliases

unambiguous semidirect productC23⋊((C37⋊C3)⋊D7)
unambiguous semidirect product(C37⋊C3)⋊D161

Splittings

Fitting, socle, derived, HallC7×C23×C37 ⋊ C2×C3
HallC7×C23×C37⋊C3 ⋊ C2
HallD5957 ⋊ C3
HallC23×C37⋊C3 ⋊ D7
HallC23×C37 ⋊ C3×D7
HallC7×C37⋊C3 ⋊ D23
HallC7×C37 ⋊ C3×D23
HallC7×C23 ⋊ D37⋊C3
HallC37⋊C3 ⋊ D161
SylowC37 ⋊ C3×D161
SylowC23 ⋊ (C37⋊C3)⋊D7
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:

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

Polycyclic presentation:

2,3,7,23,37|c^a=c^6,d^a=d^22,e^a=e^36,e^b=e^26

Conjugacy classes & character table

1212 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35742, 45);