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#72234214

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC2×H(7,𝔽2) ⋊ D3

Subgroups

subgroupG/H
center, socleC2C2×#87524
commutator, cosocle#5950C23
FrattiniC4C22×#1020
FittingC2×H(7,𝔽2)D3
hypercenterC2×D4#1020
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

7158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 7158 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(2,8)(4,17,15)(5,28,16)(7,12)(9,10,22,20,21,30)(13,24,27)(14,31,23)(18,19,29,25,26,32);(1,9,15,10)(2,4,22,5)(3,25,27,26)(6,20,28,21)(7,23,29,24)(8,16,30,17)(11,18,31,19)(12,13,32,14);(2,7)(3,11)(8,12)(9,18)(10,19)(13,23)(14,24)(20,25)(21,26)(22,29)(27,31)(30,32)

Polycyclic presentation:

2^4,3,2^5|d^2=j,b^a=bd,c^b=cj,d^a=dj,d^b=dj,e^b=e^2,f^b=g,f^e=g,g^b=f,g^e=fg,h^b=i,h^e=hi,h^f=hj,i^b=h,i^e=h,i^g=ij

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240979, #72240994, #72241002, …
maximal normal subgroup of#72240979, #72240994, #72241002, …
maximal quotient of#72343425, C2×#72234214

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