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Cayley diagram

C7×SD16 #1152

order112
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C22×C3×D4 (order 96) · Inn(G) = D4 · Out(G) = C22×C3
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×SD16

Subgroups

subgroupG/H
center, socleC2×C7D4
commutator, Frattini, cosocleC4C22×C7
prime radicalC7×D4C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD7×SD16, #6477, #6478, …
maximal subgroup ofD7×SD16, #6477, #6478, …
maximal normal subgroup ofD7×SD16, #6477, #6478, …
maximal quotient ofC7×#102, C7×#103, C7×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 25);