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C7×(C22⋊C27)⋊C2 #13094187

order1512
isoclinismisoclinic class of (C22⋊C27)⋊C2
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×(C22⋊C27)⋊C2
unambiguous semidirect product(C22⋊C27)⋊(C2×C7)

Splittings

derivedC22⋊C27 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC7(C22⋊C27)⋊C2
commutator, cosocle, hypocenterC22⋊C27C2×C7
FrattiniC9C7×S4
FittingC22×C7×C9D3
socleC22×C3×C7D9
LCSG ▷ C22⋊C27
UCSC1 ◁ C7

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 378

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3,4,5,6,7)(8,9,11,10)(13,16)(14,20)(15,23)(17,18)(19,29)(21,22)(24,31)(25,30)(26,36)(27,35)(28,34)(32,33)(37,38);(9,10,11)(12,31,25,32,36,34,14,13,35,15,38,29,21,18,17,22,19,37,23,27,16,20,28,26,33,30,24)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 84);