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(C7⋊C8)⋊A4 #10620801

order672
isoclinismisoclinic class of D7⋊A4
automorphism?
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(C7⋊C8)⋊A4
unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊C8

Splittings

derivedC22×C7 ⋊ C3×C8
HallC22×C7⋊C8 ⋊ C3
SylowC7 ⋊ C8×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC4D7⋊A4
commutator, hypocenterC22×C7C3×C8
Fitting, cosocleC22×C4×C7C2×C3
socleC23×C7C3×C4
prime radicalC2×C22⋊ΓL(1,𝔽8)C4
LCSG ▷ C22×C7
UCSC1 ◁ C4

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 168

Polycyclic presentation:

2,3,2^4,7|a^2=c,c^2=d,g^a=g^6,e^b=ef,f^b=e,g^b=g^4

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×D7⋊A4, #1131⋊A4, C2×(C7⋊C8)⋊A4, …
maximal normal subgroup ofC8×D7⋊A4, #1131⋊A4, C2×(C7⋊C8)⋊A4, …
maximal quotient of(C7⋊C8)⋊SL(2,𝔽3), (C22⋊ΓL(1,𝔽8))⋊C16, #13052179, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 383);