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C2×C3×#1914 #10639622

order768
nilpotency class3
isoclinismisoclinic class of #1913
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×C3×#1914

Subgroups

subgroupG/H
center, socleC22×C3#386
commutatorC23C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#535C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(19,20,21);(3,4,7,11,8,10,17,5)(6,13,15,18,14,9,16,12);(3,5,7,10,8,11,17,4)(6,12,16,13,14,18,15,9);(3,6)(4,9)(5,12)(7,16)(8,14)(10,13)(11,18)(15,17)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×D3×#1914, C2×#10710961, #72254312, …
maximal subgroup ofC2×C3×#12755, C3×#147817, C3×#394076, …
maximal normal subgroup ofC2×C3×#12755, C3×#147817, C3×#394076, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 14003);