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C11⋊((C67⋊C3)⋊D7) #44526276

order30954
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC11⋊((C67⋊C3)⋊D7)
unambiguous semidirect product(C67⋊C3)⋊D77

Splittings

Fitting, socle, derived, HallC7×C11×C67 ⋊ C2×C3
HallC7×C11×C67⋊C3 ⋊ C2
HallD5159 ⋊ C3
HallC11×C67⋊C3 ⋊ D7
HallC7×C67⋊C3 ⋊ D11
HallC11×C67 ⋊ C3×D7
HallC7×C67 ⋊ C3×D11
HallC67⋊C3 ⋊ D77
HallC7×C11 ⋊ D67⋊C3
SylowC67 ⋊ C3×D77
SylowC11 ⋊ (C67⋊C3)⋊D7
SylowC7 ⋊ C11⋊(D67⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11×C67C2×C3
LCSG ▷ C7×C11×C67
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 5159

A minimal faithful permutation representation — 2 generators on 85 points:

(2,3)(4,5)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(20,24,25,21,22,26)(23,32,33,27,28,42)(29,45,46,35,36,59)(30,44,40,38,34,31)(37,64,65,48,49,77)(39,62,63,51,47,56)(41,43,61,53,54,57)(50,73,72,58,67,83)(52,80,81,68,66,74)(55,60,79,69,70,75)(71,78,84,76,82,85);(1,2,4,6,7,5,3)(8,9,11,13,15,17,18,16,14,12,10)(19,20,23,29,37,50,54,28,34,42,56,72,84,82,49,66,73,75,77,81,46,61,62,45,60,76,69,53,40,25,24,30,39,52,68,51,38,22,26,31,41,55,71,70,36,47,57,59,74,65,79,67,80,64,78,85,83,63,33,44,32,43,58,48,35,27,21)

Polycyclic presentation:

2,3,7,11,67|c^a=c^6,d^a=d^10,e^a=e^66,e^b=e^37

Conjugacy classes & character table

967 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30954, 57);