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C3×#87748 #11924946

order1200
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#87748

Splittings

FittingC3×Q8×C25 ⋊ C2
HallC3×C25 ⋊ #41
SylowC25 ⋊ C3×#41

Subgroups

subgroupG/H
centerC2×C3C22×D25
commutator, cosocleC2×C25C23×C3
FrattiniC2×C5C22×C3×D5
FittingC3×Q8×C25C2
socleC2×C3×C5C22×D5
hypercenterC3×Q8D25
hypocenterC25C3×#41
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2×C3 ◁ C3×Q8

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 600

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3)(4,24,15,27,12,10,17,8,26,7,19,9,25,21,16,11,5,28,13,23,6,18,22,14,20)(29,31,32,35)(30,33,34,36);(29,30,32,34)(31,36,35,33);(5,7)(6,10)(8,13)(9,16)(11,19)(12,18)(14,15)(17,23)(20,24)(21,25)(22,27)(26,28)(31,35)(33,36)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 156);