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C3×#20301295 #36347950

order22050
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C5

Aliases

direct productC3×#20301295

Splittings

Fitting, socleC3×C52×C72 ⋊ C2×C3
derived, HallC52×C72 ⋊ C2×C32
HallC3×C7×C7⋊(C52⋊C3) ⋊ C2
HallC5⋊(C5⋊(C7⋊D7)) ⋊ C32
SylowC72 ⋊ C3×(C5⋊D5)⋊C3
SylowC52 ⋊ C3×ΓL(1,𝔽8)⋊D7

Subgroups

subgroupG/H
center, hypercenterC3#20301295
commutator, cosocle, hypocenterC52×C72C2×C32
FrattiniC1G
Fitting, socleC3×C52×C72C2×C3
LCSG ▷ C52×C72
UCSC1 ◁ C3

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 315

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,4)(3,13,9,6,7,15)(5,10,14,11,12,8)(17,20,21,18,19,22)(24,25)(26,27)(28,29);(1,3,8,14,6)(2,5,7,13,11)(4,9,10,12,15)(16,17,19,22,21,20,18)(23,24,26,28,29,27,25)(30,31,32)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22050, 387);