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Cayley diagram

C2×C4⋊C8 #422

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#347042 (order 1024) · Inn(G) = C22 · Out(G) = C22×B(4,𝔽2)
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4⋊C8

Subgroups

subgroupG/H
centerC22×C4C22
commutatorC2C22×C8
Frattini, cosocleC2×C4C23
socle, prime radicalC23C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

69 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#93221, #93222, #93223, …
Frattini subgroup of#7056, #7057, #7060, …
maximal subgroup of#1277, #1279, #1283, …
maximal normal subgroup of#1277, #1279, #1283, …
maximal quotient ofC2×#334, C2×#335, #1564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 103);