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Cayley diagram

Q8×C4⋊C4 #2350

order128
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) order 49152
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productQ8×C4⋊C4

Subgroups

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

284 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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4⋊C4×SL(2,𝔽3), #11711094, #11711095, …
maximal subgroup of#8884, #8888, #8892, …
maximal normal subgroup of#8884, #8888, #8892, …
maximal quotient ofC4⋊C42, Q8×#95, #13818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1082);