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C3×#88076 #11928066

order1248
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#88076

Splittings

HallC3×C13 ⋊ #101
SylowC13 ⋊ C3×#101

Subgroups

subgroupG/H
centerC2×C3C22⋊Dic13
commutatorC22×C13C2×C3×C4
FrattiniC2×C4C2×C3×D13
FittingC2×C3×Q8×C13C2
socleC2×C3×C13C22⋊C4
prime radicalC22×C3×C13C2×C4
cosocleC2×C4×C13C22×C3
hypercenterC2×C3×Q8D13
hypocenterC13C3×#101
LCSG ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 624

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

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

Conjugacy classes & character table

213 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 178);