Finite Group Databasebeta

← Prev (#10625405) Next (#10625407) →

C4×C7⋊H(3,𝔽3) #10625406

order756
isoclinismisoclinic class of #4427
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C32

Aliases

direct productC4×C7⋊H(3,𝔽3)
unambiguous semidirect product(C4×C7)⋊H(3,𝔽3)
unambiguous semidirect product(C32×C4)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C3×C4×ΓL(1,𝔽8))⋊C3

Splittings

FittingC32×C4×C7 ⋊ C3
HallC4×C7 ⋊ H(3,𝔽3)
SylowC7 ⋊ C4×H(3,𝔽3)

Subgroups

subgroupG/H
centerC3×C4C3×ΓL(1,𝔽8)
commutatorC3×C7C32×C4
FrattiniC2×C3C2×C3×ΓL(1,𝔽8)
FittingC32×C4×C7C3
socle, cosocleC2×C3×C7C2×C32
prime radicalC2×C7⋊H(3,𝔽3)C2
hypercenterC32×C4ΓL(1,𝔽8)
hypocenterC7C4×H(3,𝔽3)
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C3×C4 ◁ C32×C4

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 252

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C7⋊C8)⋊H(3,𝔽3), (C3⋊(C3⋊C8))⋊ΓL(1,𝔽8), #13094160, …
maximal normal subgroup of(C7⋊C8)⋊H(3,𝔽3), (C3⋊(C3⋊C8))⋊ΓL(1,𝔽8), #13094160, …
maximal quotient ofC8×C7⋊H(3,𝔽3), C2×C4×C7⋊H(3,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 37);