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Cayley diagram

C22×ΓL(1,𝔽8)⋊C4 #66660

order336
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×ΓL(1,𝔽8)⋊C4

Splittings

derived, SylowC7 ⋊ C22×C3×C4
HallC22×Dic7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23AGL(1,𝔽7)
commutator, hypocenterC7C22×C3×C4
FrattiniC2C22×AGL(1,𝔽7)
Fitting, socleC23×C7C2×C3
prime radicalC23×ΓL(1,𝔽8)C2
cosocleC2×C7C23×C3
LCSG ▷ C7
UCSC1 ◁ C23

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊#95, #6440⋊C3, #6441⋊C3, …
maximal normal subgroup ofΓL(1,𝔽8)⋊#95, #6440⋊C3, #6441⋊C3, …
maximal quotient ofC22×ΓL(1,𝔽8)⋊C8, C2×#1136⋊C3, C2×C4×ΓL(1,𝔽8)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 129);