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Cayley diagram

C2×H(3,ℤ/4) #1730

order128
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphismAut(G) order 4096
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4

Aliases

direct productC2×H(3,ℤ/4)

Subgroups

subgroupG/H
centerC2×C4C42
commutatorC4C2×C42
Frattini, cosocleC22×C4C23
socleC22#95
prime radicalC24D4
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#7254, #7256, C4×H(3,ℤ/4), …
maximal normal subgroup of#7254, #7256, C4×H(3,ℤ/4), …
maximal quotient ofC2×#1274, #7854, #7855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 462);