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Cayley diagram

C2×C4×Q8 #516

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = #72256496 (order 12288) · Inn(G) = C22 · Out(G) = #72222918
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×C4×Q8

Subgroups

subgroupG/H
centerC22×C4C22
commutatorC2C23×C4
Frattini, cosocleC22C24
socle, prime radicalC23C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22 ▷ C1

149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#142484, #142485, #142486, …
Frattini subgroup of#7206, #7238, #7248, …
Fitting subgroup ofC2×C4×SL(2,𝔽3), C2×SL(2,𝔽3)⋊C8, #73055, …
maximal subgroup of#1461, #1467, C2×Q8⋊C8, …
maximal normal subgroup of#1461, #1467, C2×Q8⋊C8, …
maximal quotient ofC2×C4×C4⋊C4, C42×Q8, C2×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 197);