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C2×H(3,𝔽5)⋊C3 #10625287

order750
isoclinismisoclinic class of H(3,𝔽5)⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC2×H(3,𝔽5)⋊C3
unambiguous semidirect product(C2×H(3,𝔽5))⋊C3

Splittings

Fitting, HallC2×H(3,𝔽5) ⋊ C3
derived, SylowH(3,𝔽5) ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC2×C5C52⋊C3
commutator, cosocle, hypocenterH(3,𝔽5)C2×C3
FrattiniC5C2×C52⋊C3
FittingC2×H(3,𝔽5)C3
LCSG ▷ H(3,𝔽5)
UCSC1 ◁ C2×C5

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 77
transitive degreeμt(G) = 150

A minimal faithful permutation representation — 2 generators on 77 points:

(1,2)(3,4,7)(5,11,27)(6,10,18)(8,19,46)(9,16,37)(12,30,47)(13,34,65)(14,20,49)(15,26,40)(17,42,66)(21,51,61)(22,48,62)(23,54,74)(24,58,32)(25,41,36)(28,45,70)(29,39,33)(31,60,73)(35,55,59)(38,52,43)(44,67,76)(50,69,64)(53,75,63)(56,71,68)(57,72,77);(3,13,12,63,42)(4,9,55,8,66)(5,33,67,70,6)(7,75,27,54,39)(10,25,30,22,28)(11,50,76,23,52)(14,64,31,61,15)(16,17,58,40,73)(18,44,49,72,69)(19,41,45,59,43)(20,24,60,57,56)(21,26,34,29,53)(32,46,74,38,37)(35,62,77,68,36)(47,71,48,65,51)

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13093712, #92525⋊C3, C4×H(3,𝔽5)⋊C3, …
maximal normal subgroup of#13093712, #92525⋊C3, C4×H(3,𝔽5)⋊C3, …
maximal quotient ofC4×H(3,𝔽5)⋊C3, C22×H(3,𝔽5)⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(750, 7);