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Cayley diagram

C7×#135 #6538

order224
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C22×C3×AGL(1,ℤ/8) (order 384) · Inn(G) = D4 · Out(G) = C24×C3
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×#135

Subgroups

subgroupG/H
centerC4×C7D4
commutator, Frattini, cosocleC4C23×C7
socleC2×C7C2×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C7 ◁ C2×C4×C7 ◁ G

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2)(3,10)(4,7)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18,19,20,21,22,23);(1,3)(2,6)(4,11)(5,13)(7,10)(8,16)(9,15)(12,14);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,15,16)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#89361, #89362, #89965, …
maximal subgroup of#89361, #89362, C7×#359, …
maximal normal subgroup of#89361, #89362, C7×#359, …
maximal quotient ofC7×#416, C7×#427, C4×C7×D8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 170);