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ΓL(1,𝔽8)×D305 #27165479

order12810
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D305
unambiguous semidirect productC5⋊(ΓL(1,𝔽8)×D61)

Splittings

Fitting, socle, derived, HallC5×C7×C61 ⋊ C2×C3
HallC5×ΓL(1,𝔽8)×C61 ⋊ C2
HallC7×D305 ⋊ C3
HallΓL(1,𝔽8)×C61 ⋊ D5
HallC7×C61 ⋊ C3×D5
HallC5×C61 ⋊ C2×ΓL(1,𝔽8)
HallC5×ΓL(1,𝔽8) ⋊ D61
SylowC61 ⋊ D5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×D61
SylowC7 ⋊ C3×D305
SylowC5 ⋊ ΓL(1,𝔽8)×D61

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C61C2×C3
LCSG ▷ C5×C7×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 73
transitive degreeμt(G) = 2135

A minimal faithful permutation representation — 2 generators on 73 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)(54,55)(56,57)(58,59)(60,61)(63,64)(65,66)(68,69,70)(71,73,72);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(62,63,65,66,64)(67,68,69,71,70,72,73)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊#11927412, ΓL(1,𝔽8)⋊#11927413, ΓL(1,𝔽8)×#11927412, …
maximal normal subgroup ofΓL(1,𝔽8)⋊#11927412, ΓL(1,𝔽8)⋊#11927413, ΓL(1,𝔽8)×#11927412, …
maximal quotient ofΓL(1,𝔽8)×Dic305, C2×ΓL(1,𝔽8)×D305, C7⋊C9×D305, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 50);