Finite Group Databasebeta

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Cayley diagram

C2×C7×M16 #6533

order224
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×C3×B(4,𝔽2) (order 768) · Inn(G) = C22 · Out(G) = C23×C3×D4
homologyAbelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22

Aliases

direct productC2×C7×M16

Subgroups

subgroupG/H
centerC2×C4×C7C22
commutatorC2C22×C4×C7
Frattini, cosocleC4C23×C7
socleC22×C7C2×C4
prime radicalC23×C7C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ G

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2)(11,12,13,14,15,16,17);(3,4,5,6,7,8,9,10);(4,8)(6,10)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13241284, #13241285, #13241286, …
Fitting subgroup of#89348, #89349, #89350, …
maximal subgroup of#89348, #89349, #89350, …
maximal normal subgroup of#89348, #89349, #89350, …
maximal quotient ofC2×C7×#97, C4×C7×M16, C2×C7×C22⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 165);