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#72227693

order1536
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

derived#73293 ⋊ C4

Subgroups

subgroupG/H
center, socleC23#5934
commutator, hypocenter#73293C4
Frattini, hypercenterC24#1020
FittingC2×#2840D3
prime radical, cosocleC2×#73293C2
LCSG ▷ #73293
UCSC1 ◁ C23 ◁ C24

853 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 853 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 28
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,3,4)(5,9,12,11,8,7,10,6)(13,19,27,25,17,26,21,18)(14,16,28,15,20,23,24,22);(6,7,12)(9,10,11)(15,16,21)(18,19,24)(22,23,27)(25,26,28)

Polycyclic presentation:

2^2,3,2^7|a^2=b,d^2=h,e^2=h,c^a=c^2,d^a=eh,d^c=e,e^a=dh,e^c=deh,e^d=eh,f^a=fg,f^c=g,f^d=fhi,f^e=fj,g^c=fg,g^d=ghij,g^e=ghi,i^a=ij

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227701

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