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C7×#859 #10620698

order672
isoclinismisoclinic class of #226
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×#859

Splittings

derivedSL(2,𝔽3) ⋊ C4×C7

Subgroups

subgroupG/H
center, socle, hypercenterC22×C7S4
commutator, hypocenterSL(2,𝔽3)C4×C7
FrattiniC22C7×S4
FittingC2×C7×Q8D3
prime radicalC2×C7×SL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C2×C7
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C7

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13057050, #13057052, #13057054, …
maximal normal subgroup of#13057050, #13057052, #13057054, …
maximal quotient ofC7×SL(2,𝔽3)⋊C8, C2×C7×#859, C7×#91886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 280);