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Cayley diagram

C7×C22⋊C8 #6415

order224
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×C3×#120 (order 384) · Inn(G) = C22 · Out(G) = C22×C3×D4
homologyAbelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C22

Aliases

direct productC7×C22⋊C8
unambiguous semidirect productC22⋊(C7×C8)

Subgroups

subgroupG/H
centerC2×C4×C7C22
commutatorC2C2×C7×C8
Frattini, cosocleC2×C4C22×C7
socleC22×C7C2×C4
prime radicalC23×C7C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ G

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 112

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#89262, #89263, #89264, …
maximal subgroup of#89262, #89263, #89264, …
maximal normal subgroup of#89262, #89263, #89264, …
maximal quotient ofC7×#323, C7×#324, C7×D4⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 47);