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Cayley diagram

C4×C4⋊Q8 #2307

order128
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 32768
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC4×C4⋊Q8

Subgroups

subgroupG/H
centerC22×C4C23
commutatorC22C23×C4
Frattini, socle, prime radical, cosocleC23C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#94129, #94133, #94141, …
maximal subgroup of#7570, #7571, #7588, …
maximal normal subgroup of#7570, #7571, #7588, …
maximal quotient ofC4×#389, C4×#391, C4×#384, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1039);