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C2×C3×#1426 #10632244

order768
nilpotency class5
isoclinismisoclinic class of #1426
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×C3×#1426

Subgroups

subgroupG/H
center, socleC22×C3#367
commutatorC16C22×C3×C4
Frattini, cosocleC2×C16C23×C3
prime radicalC23×C3SD32
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ C22×C3×C16 ◁ G

190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2)(35,36,37);(4,9,11,22)(5,14)(6,18)(7,19)(10,25)(12,26,29,34)(13,32,30,24)(15,20)(16,28)(17,27)(21,33,31,23);(3,4,10,26,6,12,5,9,19,30,14,23,15,21,25,32,8,11,28,34,20,29,17,22,7,13,27,33,18,31,16,24)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×D3×#1426, C2×#10715811
maximal subgroup ofC2×C3×#13658, C2×D3×#1426, C2×#10715811
maximal normal subgroup ofC2×C3×#13658, C2×D3×#1426, C2×#10715811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6625);