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C7×#19104885 #49141522

order36540
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×#19104885

Splittings

Fitting, socle, HallC32×C5×C7×C29 ⋊ C4
derived, HallC32×C5×C29 ⋊ C4×C7
HallC32×C7×C29 ⋊ AGL(1,𝔽5)
HallC5×C7×C29 ⋊ #157
HallC32×C5×C7 ⋊ #1181
HallC32×C29 ⋊ C7×AGL(1,𝔽5)
HallC7×C29 ⋊ #4380
HallC5×C29 ⋊ C7×#157
HallC32×C7 ⋊ #10596814
HallC32×C5 ⋊ C7×#1181
HallC5×C7 ⋊ #11760329
SylowC29 ⋊ C7×#4380
SylowC32 ⋊ C7×#10596814
SylowC5 ⋊ C7×#11760329

Subgroups

subgroupG/H
center, hypercenterC7#19104885
commutator, hypocenterC32×C5×C29C4×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C29C4
prime radicalC7×C3⋊D435C2
cosocleC3⋊D435C2×C7
LCSG ▷ C32×C5×C29
UCSC1 ◁ C7

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 47
transitive degreeμt(G) = 6090

A minimal faithful permutation representation — 2 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,12,13,11)(15,16,17,18)(20,21,22,24)(23,26,27,33)(25,30,31,29)(28,35,36,44)(32,37,39,41)(34,42,43,38)(40,45,46,47);(8,10,13)(9,12,11)(14,15,16,18,17)(19,20,23,28,37,26,34,44,47,46,39,31,24,29,38,42,30,21,25,32,40,45,35,43,33,41,36,27,22)

Conjugacy classes & character table

2310 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 364);