Finite Group Databasebeta

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Cayley diagram

D4×C7⋊C8 #89782

order448
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productD4×C7⋊C8

Splittings

SylowC7 ⋊ C8×D4

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×D7
commutatorC2×C7C22×C8
FittingC4×C7×D4C2
socleC22×C7C22×C4
prime radicalC2×C7×D4C4
cosocleC2×C4×C7C23
hypercenterC4×D4D7
hypocenterC7C8×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×D4

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

(2,3)(4,5)(6,7)(12,13,14,16,15,17,18,19);(1,2,4,6,7,5,3)(9,11);(8,9)(10,11)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11725902, #11725903, #11725904, …
maximal normal subgroup of#11725902, #11725903, #11725904, …
maximal quotient ofC22⋊C4×C7⋊C8, #11727203, #11727205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 544);