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Cayley diagram

C4×Q8 #119

order32
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×#1020 (order 384) · Inn(G) = C22 · Out(G) = C22×S4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC4×Q8

Subgroups

subgroupG/H
centerC2×C4C22
commutatorC2C22×C4
Frattini, socle, prime radical, cosocleC22C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C22 ▷ C1

35 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

derived subgroup of#3358⋊C3, #3359⋊C3, #3362⋊C3, …
Fitting subgroup ofC4×SL(2,𝔽3), SL(2,𝔽3)⋊C8, #5386, …
maximal subgroup ofQ8⋊C8, C4×SD16, C4×Q16, …
maximal normal subgroup ofQ8⋊C8, C4×SD16, C4×Q16, …
maximal quotient ofC4×C4⋊C4, #387, #389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 26);