Finite Group Databasebeta

← Prev (#34352) Next (#34354) →

Cayley diagram

C4×H(3,𝔽4) #34353

order256
nilpotency class2
isoclinismisoclinic class of #560
automorphismAut(G) order 1179648
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C42

Aliases

direct productC4×H(3,𝔽4)

Subgroups

subgroupG/H
centerC22×C4C24
commutatorC22C24×C4
Frattini, socle, cosocleC23C25
prime radicalC2×H(3,𝔽4)C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4×#5465, C4×#5450, C4×#5464, …
maximal subgroup of#97000, #129581, #129611, …
maximal normal subgroup of#97000, #129581, #129611, …
maximal quotient ofC4×#2375, C4×#2379, C4×#2670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27341);