Finite Group Databasebeta

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Cayley diagram

C4×#1161 #90264

order448
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC4×#1161

Splittings

FittingC4×C7×Q8 ⋊ C2
SylowC7 ⋊ C4×#41

Subgroups

subgroupG/H
centerC2×C4C22×D7
commutatorC2×C7C23×C4
FrattiniC22C23×D7
FittingC4×C7×Q8C2
socle, cosocleC22×C7C24
prime radicalC2×#1161C2
hypercenterC4×Q8D7
hypocenterC7C4×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×Q8

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11725983, #11725984, #11725991, …
maximal normal subgroup of#11725983, #11725984, #11725991, …
maximal quotient ofC4×#6455, C4×#6456, C4×#6461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 1026);