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Q8×C7⋊C27 #13094124

order1512
automorphism?
homologyAbelianization (H1) = C22×C27 · Schur multiplier (H2) = C1

Aliases

direct productQ8×C7⋊C27

Splittings

HallC7×Q8 ⋊ C27
SylowC7 ⋊ Q8×C27

Subgroups

subgroupG/H
center, FrattiniC2×C9C22×ΓL(1,𝔽8)
commutatorC2×C7C22×C27
FittingC7×Q8×C9C3
socle, prime radicalC2×C3×C7C22×C9
cosocleC2×C7×C9C22×C3
hypercenterQ8×C9ΓL(1,𝔽8)
hypocenterC7Q8×C27
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C9 ◁ Q8×C9

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 1512

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3,4)(5,8,7,6)(10,11,12)(13,15,14)(16,17,20,18,21,26,23,28,33,19,22,27,24,29,34,31,36,39,25,30,35,32,37,40,38,41,42);(1,5,3,7)(2,6,4,8)(9,10,11,13,12,14,15)

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1512, 21);