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C7×C61⋊C5 #14523846

order2135
isoclinismisoclinic class of C61⋊C5
automorphism?
homologyAbelianization (H1) = C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C61⋊C5
unambiguous semidirect productC61⋊(C5×C7)
unambiguous semidirect product(C7×C61)⋊C5

Splittings

Fitting, socle, HallC7×C61 ⋊ C5
derived, SylowC61 ⋊ C5×C7

Subgroups

subgroupG/H
center, hypercenterC7C61⋊C5
commutator, cosocle, hypocenterC61C5×C7
FrattiniC1G
Fitting, socleC7×C61C5
LCSG ▷ C61
UCSC1 ◁ C7

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 68
transitive degreeμt(G) = 427

A minimal faithful permutation representation — 2 generators on 68 points:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C7×C61⋊C5, C7×D61⋊C5, D7×C61⋊C5, …
maximal normal subgroup ofC2×C7×C61⋊C5, C7×D61⋊C5, D7×C61⋊C5, …
maximal quotient ofC2×C7×C61⋊C5, C49×C61⋊C5, C72×C61⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2135, 1);