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Cayley diagram

C7×C22⋊C4 #1147

order112
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C3×#120 (order 192) · Inn(G) = C22 · Out(G) = C2×C3×D4
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×C22⋊C4
unambiguous semidirect productC22⋊(C4×C7)

Subgroups

subgroupG/H
center, socleC22×C7C22
commutatorC2C2×C4×C7
Frattini, cosocleC22C22×C7
prime radicalC23×C7C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ G

34 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);(6,8)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#6380, #6440, #6441, …
maximal subgroup of#6380, C7×#99, #6440, …
maximal normal subgroup of#6380, C7×#99, #6440, …
maximal quotient ofC7×#95, C7×C22⋊C8, C7×#99, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 20);