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C2×ΓL(1,𝔽8)×C61⋊C5 #27165437

order12810
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC2×ΓL(1,𝔽8)×C61⋊C5
unambiguous semidirect product(C2×C7)⋊(C3×C61⋊C5)
unambiguous semidirect product(C2×ΓL(1,𝔽8)×C61)⋊C5

Splittings

Fitting, socle, HallC2×C7×C61 ⋊ C3×C5
derived, HallC7×C61 ⋊ C2×C3×C5
HallC2×C7×C61⋊C5 ⋊ C3
HallC2×ΓL(1,𝔽8)×C61 ⋊ C5
HallC7×C61⋊C5 ⋊ C2×C3
HallΓL(1,𝔽8)×C61 ⋊ C2×C5
HallC2×C61 ⋊ C5×ΓL(1,𝔽8)
SylowC61 ⋊ C2×C5×ΓL(1,𝔽8)
HallC2×C7 ⋊ C3×C61⋊C5
SylowC7 ⋊ C2×C3×C61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×C61⋊C5
commutator, cosocle, hypocenterC7×C61C2×C3×C5
FrattiniC1G
Fitting, socleC2×C7×C61C3×C5
LCSG ▷ C7×C61
UCSC1 ◁ C2

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 70
transitive degreeμt(G) = 854

A minimal faithful permutation representation — 2 generators on 70 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×ΓL(1,𝔽8)×C61⋊C5, ΓL(1,𝔽8)×Dic61⋊C5, ΓL(1,𝔽8)⋊C4×C61⋊C5, …
maximal normal subgroup ofC4×ΓL(1,𝔽8)×C61⋊C5, ΓL(1,𝔽8)×Dic61⋊C5, ΓL(1,𝔽8)⋊C4×C61⋊C5, …
maximal quotient ofC4×ΓL(1,𝔽8)×C61⋊C5, C22×ΓL(1,𝔽8)×C61⋊C5, C2×C7⋊C9×C61⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12810, 8);