Finite Group Databasebeta

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#72247351

order2304
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#10596770⋊C2
commutatorA4⋊S4C2×C4
Fitting, socleC25GO+(4,𝔽2)
prime radicalC2×(#1020⋊C3)⋊C2C2
cosocleC2×A4⋊S4C22
hypocenterA42C22⋊C4
LCSG ▷ A4⋊S4 ▷ A42
UCSC1 ◁ C2

530 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

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

Polycyclic presentation:

2^4,3^2,2^4|b^2=c,b^a=bd,e^a=e^2,e^b=e^2f^2,e^d=e^2,f^a=e^2f,f^d=f^2,g^a=ghi,g^e=h,g^f=h,h^a=hj,h^d=gh,h^e=gh,h^f=gh,i^a=ij,i^d=gi,i^e=ij,i^f=hi,j^b=ghij,j^d=hij,j^e=i,j^f=gj

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#10596770⋊C2
maximal normal subgroup ofC4×#10596770⋊C2

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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 4, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.7*F.9);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.9);;
gap> SetConjugate(coll, 10, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);