Finite Group Databasebeta

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Cayley diagram

C7×#117 #6520

order224
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C3×#23900 (order 1536) · Inn(G) = C22 · Out(G) = C2×C3×D42
homologyAbelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C2×C4

Aliases

direct productC7×#117

Subgroups

subgroupG/H
centerC2×C4×C7C22
commutatorC2C22×C4×C7
Frattini, cosocleC22C23×C7
socleC22×C7C23
prime radicalC23×C7C22
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ G

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#89326, #89327, #89328, …
maximal subgroup of#89326, #89327, #89328, …
maximal normal subgroup of#89326, #89327, #89328, …
maximal quotient ofC7×#376, C4×C7×C22⋊C4, C4×C7×C4⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 152);