Finite Group Databasebeta

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Cayley diagram

C7×C4⋊C4 #1148

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) = C2

Aliases

direct productC7×C4⋊C4
unambiguous semidirect productC4⋊(C4×C7)

Subgroups

subgroupG/H
center, socle, prime radicalC22×C7C22
commutatorC2C2×C4×C7
Frattini, cosocleC22C22×C7
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ G

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 112

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

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#6381, #6382, #6383, …
maximal subgroup of#6381, #6382, #6383, …
maximal normal subgroup of#6381, #6382, #6383, …
maximal quotient ofC7×#95, C7×C4⋊C8, C7×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(112, 21);