Finite Group Databasebeta

← Prev (#72246032) Next (#72246034) →

C24×ΓL(1,𝔽8)⋊D7 #72246033

order4704
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC24×ΓL(1,𝔽8)⋊D7

Splittings

Fitting, socleC24×C72 ⋊ C2×C3
derived, SylowC72 ⋊ C25×C3
HallC24×C7⋊D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C25

Subgroups

subgroupG/H
center, hypercenterC24ΓL(1,𝔽8)⋊D7
commutator, cosocle, hypocenterC72C25×C3
FrattiniC1G
Fitting, socleC24×C72C2×C3
LCSG ▷ C72
UCSC1 ◁ C24

3740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of#120×ΓL(1,𝔽8)⋊D7, C24×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246111
maximal normal subgroup of#120×ΓL(1,𝔽8)⋊D7, C24×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246111
maximal quotient ofH(5,𝔽2)×ΓL(1,𝔽8)⋊D7

GAP

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

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