Finite Group Databasebeta

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#72243199

order8400
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22×C5

Aliases

No aliases for this group.

Splittings

FittingC22×C52×C7 ⋊ C3×C4
Hall#72243086 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C22⋊C4
HallC52×C7 ⋊ C3×C22⋊C4
SylowC52 ⋊ (C22⋊Dic7)⋊C3
HallΓL(1,𝔽8) ⋊ C22⋊#1044
SylowC7 ⋊ C3×C22⋊#1044

Subgroups

subgroupG/H
center, FrattiniC2C2×ΓL(1,𝔽8)⋊#1044
commutator, socleC2×C52×C7C2×C3×C4
FittingC22×C52×C7C3×C4
prime radicalC22×ΓL(1,𝔽8)×C5⋊D5C2
cosocleC2×C7×C5⋊D5C22×C3
hypercenterC22ΓL(1,𝔽8)⋊#1044
hypocenterC52×C7C3×C22⋊C4
LCSG ▷ C2×C52×C7 ▷ C52×C7
UCSC1 ◁ C2 ◁ C22

400 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Polycyclic presentation:

2^2,3,2^2,5^2,7|a^2=d,b^a=be,f^a=f^2,f^d=f^4,g^a=f^3g^3,g^d=g^4,h^a=h^6,h^c=h^4

Conjugacy classes & character table

158 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243201, #72243219, AGL(1,𝔽7)×C22⋊#1044
maximal normal subgroup of#72243201, #72243219, AGL(1,𝔽7)×C22⋊#1044

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 5, 5, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^3*F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);