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Cayley diagram

C7×M16 #1150

order112
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×C3×D4 (order 96) · Inn(G) = C22 · Out(G) = C23×C3
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×M16

Subgroups

subgroupG/H
centerC4×C7C22
commutatorC2C2×C4×C7
Frattini, cosocleC4C22×C7
socleC2×C7C2×C4
prime radicalC22×C7C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ 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);(9,13)(11,15)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#6396, #6397, #6398, …
maximal subgroup of#6396, #6397, #6398, …
maximal normal subgroup of#6396, #6397, #6398, …
maximal quotient ofC7×#97, C7×C22⋊C8, C7×C4⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 23);