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Cayley diagram

C7×Q32 #6430

order224
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C2×C3×AGL(1,ℤ/16) (order 768) · Inn(G) = D8 · Out(G) = C22×C3×C4
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×Q32

Subgroups

subgroupG/H
center, socle, prime radicalC2×C7D8
commutator, Frattini, cosocleC8C22×C7
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×C8 ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25);(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)(13,29,21,37)(14,30,22,38)(15,31,23,39)

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#89316, #89317, D7×Q32, …
maximal subgroup of#89316, #89317, C7×SD64, …
maximal normal subgroup of#89316, #89317, C7×SD64, …
maximal quotient ofC7×#358, C7×#366, C2×C7×Q32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 62);