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Cayley diagram

C7×#106 #6423

order224
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×C3×#3003 (order 768) · Inn(G) = D4 · Out(G) = C22×C3×D4
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#106

Subgroups

subgroupG/H
center, socle, prime radicalC22×C7D4
commutatorC4C2×C4×C7
Frattini, cosocleC2×C4C22×C7
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C4×C7 ◁ G

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,4)(6,9)(7,11)(10,12)(13,14,15,16,17,18,19);(5,6,7,9,8,10,11,12)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#89282, #89283, #89624, …
maximal subgroup of#89282, #89283, C7×#360, …
maximal normal subgroup of#89282, #89283, C7×#360, …
maximal quotient ofC7×#334, C7×#340, C7×#365, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 55);