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C3×#7540 #10626147

order768
nilpotency class6
isoclinismisoclinic class of #7540
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#7540

Subgroups

subgroupG/H
center, socleC2×C3#1415
commutatorC32C2×C3×C4
Frattini, cosocleC2×C32C22×C3
prime radicalC3×#2262C2
LCSG ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ C2×C3×C32 ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 2 generators on 67 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#7540, #72255023
maximal subgroup ofC3×#153598, D3×#7540, #72255023
maximal normal subgroup ofC3×#153598, D3×#7540, #72255023

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 528);