Finite Group Databasebeta

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Cayley diagram

C2×#6467 #90432

order448
isoclinismisoclinic class of #1132
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#6467

Splittings

FittingC22×C7×C8 ⋊ C2
SylowC7 ⋊ C2×#135

Subgroups

subgroupG/H
centerC2×C4D28
commutator, cosocleC4×C7C24
FrattiniC4C23×D7
FittingC22×C7×C8C2
socleC22×C7C2×D4
hypercenterC22×C8D7
hypocenterC7C2×#135
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 4 generators on 25 points:

(3,4,6,8,9,7,5)(10,13,14,21,15,22,12,19)(11,17,16,20,18,24,23,25);(1,2)(10,12,15,14)(11,16,18,23)(13,19,22,21)(17,20,24,25);(4,5)(6,7)(8,9)(10,11)(12,16)(13,20)(14,23)(15,18)(17,21)(19,24)(22,25);(1,2)

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11726449, #11726700, #11726863, …
maximal normal subgroup of#11726449, #11726700, #11726863, …
maximal quotient ofC2×#89455, #11735887, C2×C4×D56, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1194);