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Cayley diagram

C22×H(3,ℤ/4) #20337

order256
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphismAut(G) order 196608
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC22×H(3,ℤ/4)

Subgroups

subgroupG/H
centerC22×C4C42
commutatorC4C22×C42
Frattini, cosocleC22×C4C24
socleC23#95
prime radicalC25D4
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

1135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×H(3,ℤ/4), #72341603
maximal subgroup of#123426, #123427, #123429, …
maximal normal subgroup of#123426, #123427, #123429, …
maximal quotient ofC22×#1274, C2×#7854, C2×#7855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 13325);