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Cayley diagram

C2×C4×D4 #515

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = #72225731 (order 4096) · Inn(G) = C22 · Out(G) = #72216660
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×C4×D4

Subgroups

subgroupG/H
centerC22×C4C22
commutatorC2C23×C4
Frattini, cosocleC22C24
socleC23C23
prime radicalC22×D4C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22 ▷ C1

213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 10 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#142472, #142473, #142474, …
Frattini subgroup of#7203, #7204, #7205, …
maximal subgroup of#1459, #1460, #1465, …
maximal normal subgroup of#1459, #1460, #1465, …
maximal quotient ofC2×C4×C22⋊C4, C2×C4×C4⋊C4, C42×D4, …
automorphism group of#350, C2×C5×C8, C5×M16, …
outer automorphism group ofC2×C5×C8, C2×C5×D4, C2×C3×C4×C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 196);