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ΓL(1,𝔽8)×#95 #10620533

order672
automorphism?
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C23

Aliases

direct productΓL(1,𝔽8)×#95

Splittings

Fitting, HallC7×#95 ⋊ C3
SylowC7 ⋊ C3×#95

Subgroups

subgroupG/H
center, FrattiniC23C22×ΓL(1,𝔽8)
commutatorC2×C7C3×C42
FittingC7×#95C3
socle, cosocleC23×C7C22×C3
prime radicalC23×ΓL(1,𝔽8)C22
hypercenter#95ΓL(1,𝔽8)
hypocenterC7C3×#95
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #95

152 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89246⋊C3, #89247⋊C3, ΓL(1,𝔽8)×#327, …
maximal normal subgroup of#89246⋊C3, #89247⋊C3, ΓL(1,𝔽8)×#327, …
maximal quotient ofΓL(1,𝔽8)×#336, ΓL(1,𝔽8)×H(3,ℤ/4), ΓL(1,𝔽8)×#338, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 115);