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C5×Dic77 #13095352

order1540
isoclinismisoclinic class of D77
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×Dic77
unambiguous semidirect productC7⋊(C5×Dic11)

Splittings

derived, HallC7×C11 ⋊ C4×C5
HallC5×C7×C11 ⋊ C4
HallC5×C11 ⋊ Dic7
HallC5×C7 ⋊ Dic11
SylowC11 ⋊ C5×Dic7
SylowC7 ⋊ C5×Dic11

Subgroups

subgroupG/H
center, hypercenterC2×C5D77
commutator, hypocenterC7×C11C4×C5
FrattiniC2C5×D77
Fitting, socle, prime radicalC2×C5×C7×C11C2
cosocleC2×C7×C11C2×C5
LCSG ▷ C7×C11
UCSC1 ◁ C2×C5

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 1540

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3,4,5)(6,7)(8,9)(10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27);(6,8,7,9)(11,16)(12,15)(13,14)(18,27)(19,26)(20,25)(21,24)(22,23)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC3×C5×Dic77, C5×Dic231, C5×C11⋊(ΓL(1,𝔽8)⋊C4), …
maximal normal subgroup ofC3×C5×Dic77, C5×C11⋊(ΓL(1,𝔽8)⋊C4), C5×C11×Dic77, …
maximal quotient ofC3×C5×Dic77, C5×Dic231, C5×C7⋊(C11⋊#157), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1540, 7);