Finite Group Databasebeta

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C24×C7×ΓL(1,𝔽8) #72246030

order2352
automorphism?
homologyAbelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C26

Aliases

direct productC24×C7×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC24×C72 ⋊ C3
derivedC7 ⋊ C24×C3×C7
SylowC72 ⋊ C24×C3

Subgroups

subgroupG/H
center, hypercenterC24×C7ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C24×C3×C7
FrattiniC1G
Fitting, socleC24×C72C3
LCSG ▷ C7
UCSC1 ◁ C24×C7

670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 4 generators on 22 points:

(7,8)(10,11,12)(13,15,14)(16,17,18,19,20,21,22);(5,6)(9,10,12,14,11,13,15);(3,4);(1,2)

Conjugacy classes & character table

560 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC24×ΓL(1,𝔽8)⋊D7, C7×ΓL(1,𝔽8)×#120, #72246044, …
maximal normal subgroup ofC24×ΓL(1,𝔽8)⋊D7, C7×ΓL(1,𝔽8)×#120, #72246044, …
maximal quotient ofC7×ΓL(1,𝔽8)×H(5,𝔽2)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((7,8)(10,11,12)(13,15,14)(16,17,18,19,20,21,22), (5,6)(9,10,12,14,11,13,15), (3,4), (1,2));