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Cayley diagram

C7×2−1+4 #6562

order224
nilpotency class2
isoclinismisoclinic class of H(5,𝔽2)
automorphismAut(G) = C2×C3×#11757163⋊C2 (order 11520) · Inn(G) = C24 · Out(G) = C2×C3×S5
homologyAbelianization (H1) = C24×C7 · Schur multiplier (H2) = C25

Aliases

direct productC7×2−1+4

Subgroups

subgroupG/H
center, socleC2×C7C24
commutator, Frattini, cosocleC2C24×C7
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ G

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#90017, #90018, #90528, …
maximal subgroup of#90017, #90018, C7×#454, …
maximal normal subgroup of#90017, #90018, C7×#454, …
maximal quotient ofC7×#519, C7×#520, C7×#536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 194);