#72241365

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

#62652 ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂²	C₂²×#1020
commutator, hypocenter	#997	C₂²×C₄
Fitting	#62652	D₃
prime radical	C₂²×#5934	C₂
cosocle	C₂×#997	C₂³
hypercenter	C₂²⋊C₄	#1020
LCS	G ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

3824 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 3824 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

(5,8,13,14)(6,12,18,16)(7,9,15,19)(10,17,20,11);(1,2,3,4)(6,10)(7,13,8)(11,20,12,16,18,17)(14,19,15);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241375, #72246350, C₄×#11715839, …
maximal normal subgroup of	#72241375, C₄×#11715839, #72341324, …
maximal quotient of	#72267102

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